Vector calculus, with applications to physics

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tA

VECTOR CALCULUS

WITH APPLICATIONS TO PHYSICS

BY

JAMES BYRNIE SHAW

PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF ILLINOIS

ILLUSTRATED

NEW YORK
D. VAN NOSTRAND COMPANY

Eight Warren Street
1922

s

Copyright, 1922
By D. Van Nostrand Company

All rights reserved, including that of translation into
foreign languages, including the Scandinavian

Printed in the United States of America

PREFACE.

This volume embodies the lectures given on the subject
to graduate students over a period of four repetitions. The
point of view is the result of many years of consideration
of the whole field. The author has examined the various
methods that go under the name of Vector, and finds that
for all purposes of the physicist and for most of those of the
geometer, the use of quaternions is by far the simplest in
theory and in practice. The various points of view are
mentioned in the introduction, and it is hoped that the es-
sential differences are brought out. The tables of com-
parative notation scattered through the text will assist in
following the other methods.

The place of vector work according to the author is in
the general field of associative algebra, and every method so
far proposed can be easily shown to be an imperfect form
of associative algebra. From this standpoint the various
discussions as to the fundamental principles may be under-
stood. As far as the mere notations go, there is not much
difference save in the actual characters employed. These
have assumed a somewhat national character. It is un-
fortunate that so many exist.

The attempt in this book has been to give a text to the
mathematical student on the one hand, in which every
physical term beyond mere elementary teims is carefully
defined. On the other hand for the physical student there
will be found a large collection of examples and exercises
which will show him the utility of the mathematical meth-
ods. So very little exists in the numerous treatments of
the day that does this, and so much that is labeled vector

iii

505384

IV PREFACE

analysis is merely a kind of short-hand, that it has seemed
very desirable to show clearly the actual use of vectors as
vectors. It will be rarely the case in the text that any use
of the components of vectors will be found. The triplexes
in other texts are very seldom much different fiom the ordi-
nary Cartesian forms, and not worth learning as methods.

The difficulty the author has found with other texts is
that after a few very elementary notions, the mathematical
student (and we may add the physical student) is suddenly
plunged into the profundities of mathematical physics, as
if he were familiar with them. This is rarely the case, and
the object of this text is to make him familiar with them
by easy gradations.

It is not to be expected that the book will be free from
errors, and the author will esteem it a favor to have all
errors and oversights brought to his attention. He desires
to thank specially Dr. C. F. Green, of the University of
Illinois, for his careful assistance in reading the proof, and
for other useful suggestions. Finally he has gathered his
material widely, and is in debt to many authors for it, to all
of whom he presents his thanks.

James Byrnie Shaw.
Urbana, III.,
July, 1922.

TABLE OF CONTENTS.

Chapter I. Introduction 1

Chapter II. Scalar Fields 18

Chapter III. Vector Fields 23

Chapter IV. Addition of Vectors 52

Chapter V. Vectors in a Plane 62

Chapter VI. Vectors in Space 94

Chapter VII. Applications 127

1. The Scalar of two Vectois 127

2. The Vector of two Vectors 136

3. The Scalar of three Vectors 142

4. The Vector of three Vectors 143

Chapter VIII. Differentials and Integrals 145

1. Differentiation as to one Scalar Parameter .... 145

Two Parameters 151

2. Differentiation as to a Vector 155

3. Integration 196

Chapter IX. The Linear Vector Function 218

Chapter X. Deformable Bodies 253

Strain 253

Kinematics of Displacement 265

Stress 269

Chapter XL Hydrodynamics 287

VECTOR CALCULUS

CHAPTER I
INTRODUCTION
1. Vector Calculus. By this term is meant a system of
mathematical thinking which makes use of a special class
of symbols and their combinations according to certain
given laws, to study the mathematical conclusions resulting
from data which depend upon geometric entities called
vectors, or physical entities representable by vectors, or
more generally entities of any kind which could be repre-
sented for the purposes under discussion by vectors. These
vectors may be in space of two or three or even four or
more dimensions. A geometric vector is a directed segment
of a straight line. It has length (including zero) and direc-
tion. This is equivalent to saying that it cannot be de-
fined merely by one single numerical value. Any problem
of mathematics dependent upon several variables becomes
properly a problem in vector calculus. For instance,
analytical geometry is a crude kind of vector calculus.

Several systems of vector calculus have been devised, differing in
their fundamental notions, their notation, and their laws of combining
the symbols. The lack of a uniform. notation is deplorable, but there
seems little hope of the adoption of any uniform system soon. Existing
systems have been rather ardently promoted by mathematicians of the
same nationality as their authors, and disagreement exists as to their
relative simplicity, their relative directness, and their relative logical
exactness. These disagreements arise sometimes merely with regard
to the proper manner of representing certain combinations of the
symbols, or other matters which are purely matters of convention;

1

2 YKCTOR CALCULUS

sometimes they are due to different views as to what are the import an1
things to find expressions for; and sometimes they are due to more
fundamental divergences of opinion as to the real character of the
mathematical ideas underlying any system of this sort. We will in-
dicate these differences and dispose of them in this work.

2. Bases. We may classify broadly the various systems
of vector calculus as geometric and algebraic. The former
is to be found wherever the desire is to lay emphasis on the
spatial character of the entities we are discussing, such as
the line, the point, portions of a plane, etc. The latter
lays emphasis on the purely algebraic character of the
entities with which the calculations are made, these entities
being similar to the positive and negative, and the imag-
inary of ordinary algebra. For the geometric vector
systems, the symbolism of the calculus is really nothing
more than a short-hand to enable one to follow certain
operations upon real geometric elements, with the possi-
bility kept always in mind that these entities and the
operations may at any moment be called to the front to
take the place of their short-hand representatives. For
the algebraic systems, the symbolism has to do with
hypernumbers, that is, extensions of the algebraic negative
and imaginary numbers, and does not pretend to be the
translation of actual operations which can be made visible,
any more than an ordinary calculation of algebra could be
paralleled by actual geometric or physical operations.

If these distinctions are kept in mind the different points
of view become intelligible. The best examples of geo-
metric systems are the Science of Extension of Grassmann,
with its various later forms, the Geometry of Bynames of
Study, the Geometry of Lines of Saussure, and the Geometry
of Feuillets of Cailler. The best examples of algebraic
systems are the Quaternions of Hamilton, Dyadics of Gibbs,

INTRODUCTION ,3

Multenions of McAulay, Biquaternions of Clifford, Tri-

quaternions of Combebiac, Linear Associative Algebra of

Peirce. Various modifications of these exist, and some

mixed systems may be found, which will be noted in the

proper places.

The idea of using a calculus of symbols for writing out geometric
theorems perhaps originated with Leibniz,1 though what he had in
mind had nothing to do with vector calculus in its modern sense. The
first effective algebraic vector calculus was the Quaternions of Hamilton2
(1843), the first effective geometric vector calculus was the Ausdehn-
ungslehre of Grassmann3 (1844). They had predecessors worthy of
mention and some of these will be noticed.

3. Hypernumbers. The real beginning of Vector Cal-
culus was the early attempt to extend the idea of number.
The original theory of irrational number was metric,4 and
defined irrationals by means of the segments of straight
lines. When to this was added the idea of direction, so
that the segments became directed segments, what we now
call vectors, the numbers defined were not only capable of
being irrational, but they also possessed quality, and could
be negative or positive. Ordinary algebra is thus the first
vector calculus. If we consider segments with direction
in a plane or in space of three dimensions, then we may call
the numbers they define hypernumbers. The source of the
idea was the attempt to interpret the imaginary which
had been created to furnish solutions for any quadratic or
cubic. The imaginary appears early in Cardan's work.5
For instance he gives as solution of the problem of separating
10 into two parts whose product is 40, the values
5 + V — 15, and 5 — V — 15. He considered these
numbers as impossible and of no use. Later it was dis-
covered that in the solution of the cubic by Cardan's
formula there appeared the sum of two of these impossible

4 VWCTOfl CALCULUS

values when the answer actually was real. Bombelli0 #;ive
as the solution of the cubic r3 = 15x + 4 the form

^(2 + V - 121) + ^(2 - V - 121) = 4.

These impossible numbers incited much thought and
there came about several attempts to account for them and
to interpret them. The underlying question was essen-
tially that of existence, which at that time was usually
sought for in concrete cases. The real objection to the
negative number was its inapplicability to objects. Its
use in a debit and credit account would in this sense give it
existence. Likewise the imaginary and the complex num-
ber, and later others, needed interpretation, that is, applica-
tion to physical entities.

4. Wessel, a Danish surveyor, in 1797, produced a
satisfactory method7 of defining complex numbers by means
of vectors in a plane. This same method was later given
by Argand8 and afterwards by Gauss9 in connection with
various applications. Wessel undertook to go farther and
in an analogous manner define hypernumbers by means
of directed segments, or vectors, in space of three dimen-
sions. He narrowly missed the invention of quaternions.
In 1813 Servois10 raised the question whether such vectors
might not define hypernumbers of the form

. p cos a + q cos (3 + r cos y

and inquired what kind of non-reals p, q, r would be. He
did not answer the question, however, and Wessel's paper
remained unnoticed for a century.

5. Hamilton gave the answer to the question of Servois
as the result of a long investigation of the whole problem.11
He first considered algebraic couples, that is to say in our
terminology, hypernumbers needing two ordinary numerical

INTRODUCTION 5

values to define them, and all possible modes of combining
them under certain conditions, so as to arrive at a similar
couple or hypernumber for the product. He then con-
sidered triples and sets of numbers in general. Since — 1
and i = V — 1 are roots of unity, he paid most attention
to definitions that would lead to new roots of unity.

His fundamental idea is that the couple of numbers (a, b)
where a and b are any positive or negative numbers, rational
or irrational, is an entity in itself and is therefore subject
to laws of combination just as are single numbers. For
instance, we may combine it with the other couple (x, y)
in two different ways :

(a, b) + (x, y) = (a + x, b + y)

(a, 6) X (x, y) = {ax — by, ay + bx).

In the first case we say we have, added the couples, in the
second case that we have multiplied them. It is possible
to define division also. In both cases if we set the couple
on the right hand side equal to {u, v) we find that

dujdx — dv/dy, dujdy = — dv/dx.

Pairs of functions u, v which satisfy these partial differential
equations Hamilton called conjugate functions. The partial
differential equations were first given by Cauchy in this
connection. The particular couples

€l = (1, 0), €2 = (0, 1)

play a special role in the development, for, in the first
place, any couple may be written in the form

(a, b) = aei + be2

and the notation of couples becomes superfluous; in the
second place, by defining the products of ei and e2 in various
ways we arrive at various algebras of couples. The general

C> VECTOR CALCULUS

definition would be, using the • for X,

€l'€i = Cin€i + Cii2€2, €i'€2 = Ci2i€i + ^12262,

€2'€i = C2ll€i + C212€2> «2 * €2 = C221«l + C222€2-

By varying the choice of the arbitrary constants c, and
Hamilton considered several different cases, different
algebras of couples could be produced. In the case above
the c's are all zero except

Cm = 1, C122 = 1, C212 — 1, C221 = — 1.

From the character of 4 it may be regarded as entirely
identical with ordinary 1, and it follows therefore that
e2 may be regarded as identical with the V — 1. On the
other hand we may consider €1 to be a unit vector pointing
to the right in the plane of vectors, and c2 to be a unit
vector perpendicular to ei. We have then a vector calculus
practically identical with Wessel's. The great merit of
Hamilton's investigation lies of course in its generality.
He continued the study of couples by a similar study of
triples and then quadruples, arriving thus at Quaternions.
His chief difference in point of view from those who followed
him and who used the concept of couple, triple, etc. {Mul-
tiple we will say for the general case), is that he invariably
defined one product, whereas others define usually several.
6. Multiples. There is a considerable tendency in the
current literature of vector calculus to use the notion of
multiple. A vector is usually designated by a triple as
(x, y, z), and usually such triple is called a vector. It is
generally tacitly understood that the dimensions of the
numbers of the triple are the same, and in fact most of the
products defined would have no meaning unless this
homogeneity of dimension were assumed to hold. We
find products defined arbitrarily in several ways. For
instance, the scalar product of the triples (a, b, c) and (x, y, z)

INTRODUCTION 7

is =fc (ax + by + cz), the sign depending upon the person
giving the definition; the vector product of the same two
triples is usually given as the triple (bz — cy, ex — az,
ay — bx). It is obvious at once that a great defect of such
definitions is that the triples involved have no sense until
the significance of the first number, the second number,
and the third number in each triple is understood. If
these depend upon axes for their meaning, then the whole
calculus is tied down to such axes, unless, as is usually
done, the expressions used in the definitions are so chosen
as to be in some respects independent of the particular
set of axes chosen. When these expressions are thus
chosen as invariants under given transformations of the
axes we arrive at certain of the well-known systems of
vector analysis. The transformations usually selected to
furnish the profitable expressions are the group of orthog-
onal transformations. For instance, it was shown by
Burkhardt12 that all the invariant expressions or invariant
triples are combinations of the three following :

ax + by + cz,

(bz — cy, ex — az, ay — bx),

(al + bm + cn)x + (am — bl)y + (an — cl)z,

(bl — am)x + (al -f- bm + cn)y + (bn — cm)z,

(cl — ari)x + (cm — bn)u + (al + bm + cn)z.

A study of vector systems from this point of view has
been made by Schouten.13

7. Quaternions. In his first investigations, Hamilton
was chiefly concerned with the creation of systems of
hypernumbers such that each of the defining units, similar
to the ei and €2 above, was a root of unity.14 That is, the
process of multiplication by iteration would bring back the
multiplicand. He was actually interested in certain special

8 VECTOR CALCULUS

cases of abstract groups,15 and if he had noticed the group
property his researches would perhaps have extended into
the whole field of abstract groups. In quaternions he found
a set of square roots of — 1, which he designated by i, j, k,
connected with his triples though belonging to a set of quad-
ruples. In his Lectures on Quaternions, the first treatise he
published on the subject, he chose a geometrical method of
exposition, consequently many have been led to think of
quaternions as having a geometric origin. However, the
original memoirs show that they were reached in a purely
algebraic way, and indeed according to Hamilton's philoso-
phy were based on steps of time as opposed to geometric
steps or vectors.

The geometric definition is quite simple, however, and
not so abstract as the purely algebraic definition. Ac-
cording to this idea, numbers have a metric definition, a
number, or hypernumber, being the ratio of two vectors.
If the vectors have the same direction we arrive at the
ordinary numerical scale. If they are opposite we arrive
at the negative numbers. If neither in the same direction
nor opposite we have a more general kind of number, a
hypernumber in fact, which is a quaternion, and of which
the ordinary numbers and the negative numbers are
merely special cases. If we agree to consider all vectors
which are parallel and in the same direction as equivalent,
that is, call them free vectors, then for every pair of vectors
from the origin or any fixed point, there is a quaternion.
Among these quaternions relations will exist, which will
be one of the objects of study of later chapters.

8. Mobius was one of the early inventors of a vector
calculus on the geometric basis. In his Barycentrisch.es
Kalkul16 he introduced a method of deriving points from
other points by a process called addition, and several

INTRODUCTION 9

applications were made to geometry. The barycentric
calculus is somewhat between a system of homogeneous
coordinates and a real vector calculus. His addition was
used by Grassmann.

9. Grassmann in 1844 published his treatise called Die
lineale Ausdehnungslehre17 in which several different proc-
esses called multiplication are used for the derivation of
geometric entities from other geometric entities. These
processes make use of a notation which is practically a
sort of short-hand for the geometric processes involved.
Grassmann considered these various kinds of multiplication
abstractly, leaving out of account the meaning of the
elements multiplied. His methods apply to space of N
dimensions. In the symmetric multiplication it is possible
to interchange any two of the factors without affecting the
result. In the circular multiplication the order may be
changed cyclically. In the lineal multiplication all the
laws hold as well for any factors which are linear combina-
tions of the hypernumbers which define the base, as for
those called the base. He studies two species of circular
multiplication. If the defining units of the base are ex, e2, e3
• • •€„, then we have in the first variety of circular multipli-
cation the laws

€l2 + €22 + 632 + • • • + €n2 = 0, €i€j = €j€i.

In the second variety we have the laws

ei2 = 0, e/ = 0, • - • en2 = 0, Mi = 0, *+j.

In the lineal genus of multiplication he studies two
species, in the first, called the algebraic multiplication, we
have the law

My = *fii for all i, j.

while in the second, called the exterior multiplication, the
interchange of any two factors changes the sign of the

10 VECTOR CALCULUS

result. Of the latter there are two varieties, the progressive
multiplication in which the number of dimensions of the
geometric figure which is the product is the sum of the
dimensions of the factors, while in the other, called re-
gressive multiplication, the dimension of the product is the
difference between the sum of the dimensions of the factors
and N the dimension of the space in which the operation
takes place. From the two varieties he deduces another
kind called interior multiplication.

If we confine our thoughts to space of three dimensions,
defined by points, and if €1, e2, e3, e4 are such points, the
progressive exterior product of two, as €1, e2, is ei€2 and
represents the segment joining them if they do not coincide.
The product is zero if they coincide. The product of this
into a third point €3 is ei€2e3 and represents the parallelogram
with edges €162, ei€3 and the other two parallel to these
respectively. If all three points are in a straight line the
product is zero. The exterior progressive product c1e2e3€4
represents the parallelepiped with edges €ie2, €ie3, €i€4 and
the opposite parallel edges. The regressive exterior product
of €i€2 and €ie3€4 is their common point €1. The regressive
product of €ie2e3 and €ie2€4 is their common line €ie2. The
complement of €1 is defined to be €2e3e4, and of €i€2 is e3fct,
and of €i€2e3 is €4. The interior product of any expression
and another is the progressive or regressive product of the
first into the complement of the other. For instance, the
interior product of €1 and e2 is the progressive product of
€1 and €i€3e4 which vanishes. The interior product of e2
and e2 is the product of e2 and eie3e4 which is €2eie3e4. The
interior product of €j€2e3 and ei€4 is the product of €ie2e3
and €2e3 which would be regressive and be the line e2e3.

We have the same kinds of multiplication if the expres-
sions e are vectors and not points, and they may even be

INTRODUCTION 1 1

planes. The interpretation is different, however. It is
easy to see that Grassmann's ideas do not lend themselves
readily to numerical application, as they are more closely
related to the projective transformations of space. In
fact, when translated, most of the expressions would be
phrased in terms of intersections, points, lines and planes,
rather than in terms of distances, angles, areas, etc.

10. Dyadics were invented by Gibbs,18 and are of both the
algebraic and the geometric character. Gibbs has, like
Hamilton, but one kind of multiplication. If we have
given two vectors a, (3 from the same point, their dyad is a(3.
This is to be looked upon as a new entity of two dimensions
belonging to the point from which the vectors are drawn.
It is not a plane though it has two dimensions, but is really
a particular and special kind of dyadic, an entity of two-
dimensional character, such that in every case it can be
considered to be the sum of not more than three dyads.
Gibbs never laid any stress on the geometric existence of
the dyadic, though he stated definitely that it was to be
considered as a quantity. His greatest stress, however,
was upon the operative character of the dyadic, its various
combinations with vectors being easily interpretable. The
simplest interpretation is from its use in physics to represent
strain.

Gibbs also pushed his vector calculus into space of many
dimensions, and into triadic and higher forms, most of
which can be used in the theory of the elasticity of crystals.
The scalar and vector multiplication he considered as
functions of the dyadic, rather than as multiplications,
and there are corresponding functions of triadics and
higher forms. In this respect his point of view is close to
that of Hamilton, the difference being in the use of the
dyadic or the quaternion.

11. Other forms of vector calculus can be reduced to
3

12 VECTOR CALCULUS

these or to combinations of parts of these. The differences
are usually in the notations, or in the basis of exposition.

Notations for One Vector

Greek letters, Hamilton, Tait, Joly, Gibbs.

Italics, Grassmann,_Peano, Fehr, Ferraris, Macfarlane.
Heun writes a, b, c.

Old English or German letters, Maxwell, Jaumann, Jung,
Foppl, Lorentz, Gans, Abraham, Bucherer, Fischer,
Sommerfeld.

Clarendon type, Heaviside, Gibbs, Wilson, Jahnke, Timer-
ding, Burali-Forti, Marcolongo.

Length of a vector
T ( ), Hamilton, Tait, Joly.
| | , Gans, Bucherer, Timerding.
Italic corresponding to the ve ctor letter, Wilson, Jaumann,

&ing, Fischer, Jahnke. Corresponding small italic,

Macfarlane.
Mod. ( ), Peano, Burali-Forti, Marcolongo, Fehr.

Unit of a vector
U ( ), Hamilton, Tait, Joly, Peano.
Clarendon small, Wilson.
( )i, Bucherer, Fischer.
Corresponding Greek letter, Macfarlane.
Some write the vector over the length.

Square of a vector
( )2. The square is usually positive except in Quaternions,
where it is negative.

Reciprocal

( )-1, Hamilton, Tait, Joly, Jaumann.

tt , Hamilton, Tait, Joly, Fischer, Bucherer.

CHAPTER II

SCALAR FIELDS
1. Fields. If we consider a given set of elements in
space, we may have for each element one or more quantities
determined, which can be properly called functions of the
element. For instance, at each point in space we may have
a temperature, or a pressure, or a density, as of the air.
Or for every loop that we may draw in a given space we
may have a length, or at some fixed point a potential due
to the loop. Again, we may have at each point in space
a velocity which has both direction and length, or an
electric intensity, or a magnetic intensity. Not to multiply
examples unnecessarily, we can see that for a given range
of points, or lines, or other geometric elements, we may
have a set of quantities, corresponding to the various
elements of the range, and therefore constituting a function
of the range, and these quantities may consist of numerical
values, or of vectors, or of other hypernumbers. When
they are of a simple numerical character they are called
scalars, and the function resulting is a scalar function.
Examples are the density of a fluid at each point, the density
of a distribution of energy, and similar quantities consisting
of an amount of some entity per cubic centimeter, or per
square centimeter, or per centimeter.

EXAMPLES
(1) Electricity. The unit of electricity is the coulomb,
connected with the absolute units by the equations
1 coulomb = 3 • 10° electrostatic units

== 10-1 electromagnetic units.
13

14 VECTOR CALCULUS

The density of electricity is its amount in a given volume,
area, op length divided by the volume, area, or length
respectively. The dimensions of electricity will be repre-
sented by [9], and for its amount the symbol 9 will be used.
For the volume density we will use e, for areal density e' ,
for linear density e". If the distribution may be considered
to be continuous, we may take the limits and find the
density at a point.

(2) Magnetism. Considering magnetism to be a quan-
tity, we will use for the unit of measurement the maxwell,
connected with the absolute units by the equation

1 maxwell = 3-1010 electrostatic units

= 1 electromagnetic unit.

Sometimes 108 maxwells is called a weber. The symbol for
magnetism will be $, the dimensions [$], the densities
m, m', m".

(3) Action. This quantity is much used in physics, the
principle of least action being one of the most important
fundamental bases of modern physics. The dimensions
of action are [93>], the symbol we shall use is A, and the
unit might be a quantum, but for practical purposes a
joule-second is used. In the case of a moving particle the
action at any point depends upon the path by which the
particle has reached the point, so that as a function of the
points of space it has at each point an infinity of values.
A function which has but a single value at a point will be
called monodromic, but if it has more than one value it will
be called polydromic. The action is therefore a polydromic
function. We not only have action in the motion of par-
ticles but we find it as a necessary function of a momentum
field, or of an electromagnetic field.

(4) Energy. The unit of energy is the erg or the joule

SCALAR FIELDS 15

= 107 ergs. Its dimensions are [G^T7-1], its symbol will
beW.

(5) Activity. This should not be confused with action.
It is measured in watts, symbol J, dimensions [Q$T~2].

(6) Energy-density. The symbol will be U, dimensions

(7) Activity-density. The symbol will be Q, dimensions

pi-3r2].

(8) Mass. The symbol is M, dimensions [0$77r2].
The unit of mass is the gram. A distribution of mass is
usually called a distribution of matter.

(9) Density of mass. The symbol will be c, dimensions

(10) Potential of electricity. Symbol V, dimensions

(11) Potential of magnetism. Symbol N, dimensions
[027-1].

(12) Potential of gravity. Symbol P, dimensions [G^T7-1].

2. Levels. Points at which the function has the same
value, are said to define a level surface of the function. It
may have one or more sheets. Such surfaces are usually
named by the use of the prefixes iso and equi. For instance,
the surfaces in a cloud, which have all points at the same
temperature, are called isothermal surfaces; surfaces which
have points at the same pressure are called isobaric surfaces;
surfaces of equal density are isopycnic surfaces; those of
equal specific volume (reciprocal of the density) are the iso-
steric surfaces; those of equal humidity are isohydric surfaces.
Likewise for gravity, electricity, and magnetism we have
equipotential surfaces.

3. Lamellae. Surfaces are frequently considered for
which we have unit difference between the values of the
function for the successive surfaces. These surfaces and

16 VECTOR CALCULUS

the space between them constitute a succession of unit
lamellae.

If we follow a line from a point A to a point B, the number
of unit lamellae traversed will give the difference between
the two values of the function at the points A and B.
If this is divided by the length of the path we shall have the
mean rate of change of the function along the path. If
the path is straight and the unit determining the lamellae
is made to decrease indefinitely, the limit of this quotient
at any point is called the derivative of the function at
that point in the given direction. The derivative is ap-
proximately the number of unit lamellae traversed in a
unit distance, if they are close together.

4. Geometric Properties. Monodromic levels cannot in-
tersect each other, though any one may intersect itself.
Any one or all of the levels may have nodal lines, conical
points, pinch-points, and the other peculiarities of geo-
metric surfaces. These singularities usually depend upon
the singularities of the congruence of normals to the
surface.

In the case of functions of two variables, the scalar levels
will be curves on the surface over which the two variables
are defined. Their singularities may be any that can
occur in curves on surfaces.

5. Gradient. The equation of a level surface is found
by setting the function equal to a constant. If, for in-
stance, the point is located by the coordinates x, y, z
and the function is f(x, y, z), then the equation of any
level is

u = /(*> V> z) = C.

If we pass to a neighboring point on the same surface
we have

du = f{x + dx, y -f- dy, z + dz) — f{x, y, z) = 0.

We may usually find functions df/dx, bf\a\ df/dz,

SCALAR FIELDS 17

functions independent of dx, dy, dz, such that

du — dfjdx • dx + df/dy • dy + df/dz • dz.

Now the vector from the first point to the second has
as the lengths of its projections on the axes: dx, dy, dz; and
if we define a vector whose projections are dfjdx, df/dy,
df/dz, which we will call the Gradient of f, then the con-
dition du = 0 is the condition that the gradient of / shall be
perpendicular to the differential on the surface. Hence,
if we represent the gradient of / by v/, and the differential
change from one point to the other by dp, we see that dp
is any infinitesimal tangent on the surface and v/ is along the
normal to the surface. It is easy to see that if we differen-
tiate u in a direction not tangent to a level surface of u we
shall have

du = df/dx-dx + df/dy •<&,+ df/dz -dz = dC.

If the length of the differential path is ds then we shall have*
du/ds = projection of^fon the unit vector in the direction of dp.
The length of the vector v/ is sometimes called the gradient
rather than the vector itself. Sometimes the negative of
the expression used here is called the gradient.

When the three partial derivatives of / vanish for the
same point, the intensity of the gradient, measured by its
length, is zero, and the direction becomes indeterminate
from the first differentials. At such points there are singu-
larities of the function. At points where the function
becomes infinite, the gradient becomes indeterminate and
such points are also singular points.

6. Potentials. The three components of a vector at a
point may be the three partial derivatives of the same
function as to the coordinates, in which case the vector
may be looked upon as the gradient of the integral func-

* Since dxjds, dyjds, dzjds are the direction-cosines of dp.

18 VECTOR CALCULUS

tion, which is called a potential junction, or sometimes a
force function. For instance, if the components of the
velocity satisfy the proper conditions, the velocity is the
gradient of a velocity 'potential. These conditions will be
discussed later, and the vector will be freed from dependence
upon any axes.

7. Relative Derivatives. In case there are two scalar
functions at a point, we may have use for the concept of
the derivative of one with respect to the other. This is
defined to be the quotient of the intensity of the gradient of
the first by that of the second, multiplied by the cosine
of their included angle. If the unit lamellae are constructed,
it is easy to see from the definition that the relative deriva-
tive of the first as to the second will be the limit of the
average or mean of the number of unit sheets of the first
traversed from one point to another, along the normal of the
second divided by the number of unit sheets of the second
traversed at the same time. For instance, if we draw the
isobars for a given region of the United States and the
simultaneous isotherms, then in passing from a point A
to a point B if we traverse 24 isobaric unit sheets and 10
isothermal unit sheets, the average is 2.4 isobars per
isotherm. ^

8. Unit-Tubes. If there are two scalar functions in the
field, and the unit lamellae are drawn, the unit sheets will
usually intersect so as to divide the space under considera-
tion into tubes whose cross-section will be a curvilinear
parallelogram. Since the area of such parallelogram is
approximately

dsids2 esc 0,
where dsi is the distance from a unit sheet of the function u
to the next unit sheet, and ds2 the corresponding distance
for the function v, while 6 is the angle between the surfaces;
and since we have, Tyu being the intensity of the gradient

SCALAR FIELDS 19

of u, and T^/v the intensity of the gradient of v,

dsi - 1/TVu, ds2 = 1/Tw

the area of the parallelogram will be l/(TyuTvv sin 6).
Consequently if we count the parallelograms in any plane

Fig. 1.

cross-section of the two sets of level surfaces, this number
is an approximate value of the expression

T^uT^Jv sin 6 X area parallelogram

when summed over the plane cross-section. That is to
say, the number of these tubes which stand perpendicular
to the plane cross-section is the approximate integral of the
expression T^uT^v sin 6 over the area of the cross-section.
These tubes are called unit tubes for the same reason that
the lamellae are called unit lamellae.

In counting the tubes it must be noticed whether the
successive surfaces crossed correspond to an increasing or
to a decreasing value of u or of v. It is also clear that
when sin 6 is everywhere 0 the integral must be zero. In
such case the three Jacobians

d(u, v)/d(y, z), d(u, v)/d(z, x), d{u, v)/d(x, y)

20 VECTOR CALCULUS

are each equal zero, and this is the^condition that u is a
function of v. In case the plane of cross-section is the
x, y plane, the first two expressions vanish anyhow, since
u, v are functions of x, y only.

It is clear if we take the levels of one of the functions,
say u, as the upper and lower parts of the boundary of the
cross-section, that in passing from one of the other sides
of the boundary along each level of u the number of unit
tubes we encounter from that side of the boundary to the
opposite side is the excess of the value of v on the second
side over that on the first side. If then we count the dif-
ferent tubes in the successive lamellae of u between the
two sides of the cross-section we shall have the total excess
of those on the second side over those on the first side.
That is to say, the number of unit tubes or the integral
over the area bounded by level 1 and level 2 of u, and any
other two lines which cross these two levels so as to produce
a simple area between, is the excess of the sum between
the two levels of the values of v on one side over the same
sum between the two levels of u on the other side. These
graphical solutions are used in Meteorology.

This gives the excess of the integral J vdu along the

second line between the two levels of u, over the same in-
tegral along the first line. It represents the increase of this
integral in a change of path from one line to the other. For
instance if the integral is energy, the number of tubes is
the amount of energy stored or released in the passage from
one line to the other, as in a cyclone. The number of tubes

for any closed path is the approximate integral I rdu

around the path. ,

SCALAR FIELDS 21

EXERCISES.

1. If the density varies as the distance from a given axis, what are
the isopycnic surfaces?

2. A rotating fluid mass is in equilibrium under the force of gravity,
the hydrostatic pressure, and the centrifugal force. What are the
levels? Show that the field of force is conservative.

3. The isobaric surfaces are parallel planes, and the isopycnic
surfaces are parallel planes at an angle of 10° with the isobaric planes.
What is the rate of change of pressure per unit rate of change of density
along a line at 45° with the isobaric planes?

4. If the pressure can be stated as a function of the density, what
conditions are necessary? Are they sufficient? What is the interpreta-
tion with regard to the levels?

5. Three scalar functions have a functional relation if their Jacobian
vanishes. What does this mean with regard to their respective levels?

6. If the isothermal surfaces are spheres with center at the earth's
center, the temperature sheets for decrease of one degree being 166.66
feet apart, and if the isobaric levels are similar spheres, the pressure
being given by

log B = log B, - 0.0000177 (a - z0),

where B0 is the pressure at z0 feet above the surface of the earth, what
is the relative derivative of the temperature as to the pressure, and the
pressure as to the temperature?

7. To find the maximum of u(x, y, z) we set du = 0. If there is also
a condition to be fulfilled, v(x, y, z) = 0, then dv = 0 also.

These two equations in dx, dy, dz must be satisfied for all compatible
values of dx, dy, dz, and we must therefore have

du du du _ _ dy # dv dv_
dx' dy' dz' ~ dx' dy' dz}

which is equivalent to the single vector equation

Vw = wyv.

What does this mean in terms of the levels :; The unit tubes?

If there is also another equation of condition l(x, y, z) =0 then also
dt = 0 and the Jacobian of the three functions u, v, t must equal zero.
Interpret.

8. On the line of intersection of two levels of two different functions
the values of both functions remain constant. If we differentiate a
third function along the locus in question, the differential vanishing
everywhere, what is the significance?

22

VECTOR CALCULUS

9. If a field of force has a potential, then a fluid, subject to the force
and such that its pressure is a function of the density and the tempera-
ture, will have the equipotential levels for isobaric levels also. The
density will be the derivative of the pressure relative to the potential.
Show therefore that equilibrium is not possible unless the isothermals
are also the levels of force and of pressure.

[p = p(c, T), and vp = cvv = PcVc + prvT.

If then vc = 0, cvv = prVT.]

10. If the full lines below represent the profiles of isobaric sheets, and
the dotted lines the profiles of isosteric sheets, count the unit tubes
between the two verticals, and explain what the number means. If
they were equipotentials of gravity and isopycnic surfaces, what would
the number of unit tubes mean?

Fig. 2.

11. If u = y — 12x3 and v = y + x2 + \x, find Vw and w and
TvuTw -sin 6, and integrate the latter over the area between x = 0f
x = 1, y = 0, y = 12. Draw the lines.

12. If u = ax + by + cz and v = x2 -f- if + z2, find vw and vv and
TyuTvvsm 6 and integrate the latter expression over the surface of a
cylinder whose axis is in the direction of the z axis. Find the deriva-
tive of each relative to the other.

CHAPTER III

VECTOR FIELDS

1. Hypercomplex Quantity. In the measurement of
quantity the first and most natural invention of the mind
was the ordinary system of integers. Following this came
the invention of fractions, then of irrational numbers.
With these the necessary list of numbers for mere measure-
ment of similar quantities is closed, up to the present time.
Whether it will be necessary to invent a further extension
of number along this line remains for the future to show.

In the attempt to solve equations involving ordinary
numbers, it became necessary to invent negative numbers
and imaginary numbers. These were known and used as
fictitious numbers before it was noticed that quantities
also are of a negative or an "imaginary" character. We
find instances everywhere. In debit and credit, for ex-
ample, we have quantity which may be looked upon as of
two different kinds, like iron and time, but the most logical
conception is to classify debits and credits together in the
single class balance. One's balance is what he is worth
when the debits and credits have been compared. If the
preponderance is on the side of debit we consider the balance
negative, if on the side of credit we consider the balance
positive. Likewise, we may consider motion in each direc-
tion of the compass as in a class by itself, never using any
conception of measurement save the purely numerical one
of comparing things which are exactly of the same kind
together. But it is more logical, and certainly more general,
to consider motions in all directions of the compass and
of any distances as all belonging to a single class of quantity.

23

24 VECTOR CALCULUS

In that case the comparison of the different motions leads
us to the notion of complex numbers. When Wessel made
his study of the vectors in a plane he was studying the
hypernumbers we usually call "the complex field." The
hypernumbers had been studied in themselves before, but
were looked upon (rightly) as being creations of the mind
and (in that sense correctly) as having no existence in what
might be called the real world. However, their deduction
from the vectors in a plane showed that they were present
as relations of quantities which could be considered as alike.
Again when Steinmetz made use of them in the study of
the relations of alternating currents and electromotive
forces, it became evident that the so-called power current
and wattless current could be regarded as parts of a single
complex current, and similarly for the electromotive forces.
The laws of Ohm and Kirchoff could then be generalized so
as to be true for the new complex quantities. In this brief
history we find an example of the interaction of the develop-
ments of mathematics. The inventions of mathematics
find instances in natural phenomena, and in some cases
furnish new conceptions by which natural phenomena can
be regarded as containing elements that would ordinarily
be completely overlooked.

In space of three (or more) dimensions, the vectors
issuing from a point in all directions and of all lengths
furnish quantities which may be considered to be all of
the same kind, on one basis of classification. Therefore,
they will define certain ratios or relations which may be
called hypernumbers. This is the class of hypernumbers
we are particularly concerned with, though we shall occa-
sionally notice others. Further, any kind of quantity
which can be represented completely for certain purposes
by vectors issuing from a point we will call vector quantity.

VECTOR FIELDS 25

Such quantities, for instance, are motions, velocities,
accelerations, at least in the Newtonian mechanics, forces,
momenta, and many others. The object of VECTOR CAL-
CULUS is to study these hypernumbers in relation to their
corresponding quantities, and to derive an algebra capable
of handling them.

We do not consider a vector as a mere triplex of ordinary numbers.
Indeed, we shall consider two vectors to be identical when they
represent or can represent the same quantity, even though one is ex-
pressed by a certain triplex, as ordinary Cartesian coordinates, and the
other by another triplex, as polar coordinates. The numerical method
of defining the vector will be considered as incidental.

2. Notation. We shall represent vectors for the most
part by Greek small letters. Occasionally, however, as
in Electricity, it will be more convenient to use the standard
symbols, which are generally Gothic type. As indicated
on page 12 there is a great variety of notation, and only
one principle seems to be used by most writers, namely
that of using heavy type for vectors, whatever the style of
type. In case the vector is from the origin to the point
(x, y, z) it may be indicated by

Px, y, z>

while for the same point given by polar coordinates r, <p, 6
we may use

Pr, <p, 6)

In case a vector is given by its components as X, Y, Z we
will indicate it by

?x, y, z

3. Equivalence. All vectors which have the same direc-
tion and same length will be considered to be equivalent.
Such vectors are sometimes called free vectors. The term
vector will be used throughout this book, however, with no
other meaning.

2G VECTOR CALCULUS

In case vectors are equivalent only when they lie on the
same line, and have the same direction and length, they
will be called glissants. A force applied to a rigid body
must be considered to be a glissant, not a vector. In
case vectors are equivalent only when they start at the
same point and coincide, they will be called radials. The
resultant moment of a system of glissants with respect to a
point A is a radial from A.

The equivalence of two vectors

a = 0

implies the existence of equalities infinite in number, for
their projections on any other lines will then be equal. The
infinite set of equalities, however, is reducible in an infinity
of ways to three independent equalities. For instance, we
may write either
ax = ft., ay = fiy, a2 = 13 z, or ar = Br, a<p = ^lf>,alf! = /?„.

The equivalence of two glissants implies sets of equalities
reducible in every case to five independent equalities. The
equivalence of two radials reduces to sets of six equalities.

4. Vector Fields. Closely allied to the notion of radial

is that of vector field. A vector field is a system of vectors

each associated with a point of space, or a point of a surface,

or a point of a line or curve. The vector is a function of

the position of the point which is itself usually given by a

vector, as p. The vector function may be monodromic or

polydromic. We will consider some of the usual vector

fields.

EXAMPLES

(1) Radius Vector, p [L]. This will usually be indicated
by p. In case it is a function of a single parameter, as t,
the points defined will lie on a curve;* in case it is a function

* We are discussing mainly ordinary functions, not the "pathologic
type."

VECTOR FIELDS 27

of two parameters, u, v, the points defined will lie on a
surface. The term vector was first introduced by Hamilton
in this sense. When we say that the field is p, we mean
that at the point whose vector is p measured from the fixed
origin, there is a field of velocity, or force, or other quantity,
whose value at the point is p.

(2) Velocity, a [XT7-1]. Usually we will designate veloc-
ity by c. In the case of a moving gas or cloud, each particle
has at each point of its path a definite velocity, so that we
can describe the entire configuration of the moving mass at
any instant by stating what function a is of p, that is, for
the point at the end of the radius vector p assign the velocity
vector. The path of a moving particle will be called a
trajectory. At each point of the path the velocity a is a
tangent of the trajectory.

If we lay off from a fixed point the vectors a which corre-
spond to a given trajectory, their terminal points will
lie on a locus called by Hamilton the hodograph of the
trajectory. For instance, the hodographs of the orbits of
the planets are circles, to a first approximation. If we
multiply a by dt, which gives it the dimensions of length,
namely an infinitesimal length along the tangent of the
trajectory, the differential equation of the trajectory
becomes

dp = adt.

The integral of this in terms of t gives the equation of the
trajectory.

(3) Acceleration. t[LT~2]. An acceleration field is simi-
lar to a velocity field except in dimensions. The accelera-
tion is the rate of change of the vector velocity at a point,
consequently, if a point describes the hodograph of a trajec-
tory so that its radius vector at a given time is the velocity
in the trajectory at that time, the acceleration will be a

3

L\S VECTOR CALCULUS

tangent to the hodograph, and its length will be the velocity
of the moving point in the hodograph. We will use r to
indicate acceleration.

(4) Momentum Density. T [$QL~4]. This is a vector
function of points in space and of some number which can
be attached to the point, called density. In the case of a
moving cloud, for instance, each point of the cloud will have
a velocity and a density. The product of these two factors
will be a vector whose direction is that of the velocity and
whose length is the product of the length of the velocity
vector and the density. However, momentum density
may exist without matter and without motion. In electro-
dynamic fields, such as could exist in the very simple case
of a single point charge of electricity and a single magnet
pole at a point, we also have at every point of space a
momentum density vector. This may be ascribed to the
hypothetical motion of a hypothetical ether, but the essen-
tial feature is the existence of the field. If we calculate the
integral of the projection of the momentum density on the
tangent to a given curve from a point A to a point B, the
value of the integral is the action of an infinitesimal volume,
an action density, along that path from A to B. The
integration over a given volume would give the total
action for all the particles over their various paths. This
would be a minimum for the paths actually described as
compared with possible paths. Specific momentum is
momentum density of a moving mass.

(5) Momentum. Y [TOL-1]. The volume integral of
momentum density or specific momentum is momentum.
Action is the line-integral of momentum.

(6) Force Density. F [^QL^T-1]. If a field of momen-
tum density is varying in time then at each point there is a
vector which may be called force-density, the time derivative

VECTOR FIELDS 29

of the momentum density. Such cases occur in fields due
to moving electrons or in the action of a field of electric
intensity upon electric density, or magnetic intensity on
magnetic density.

(7) Force. X [mL-1?7-1]. The unit of force has re-
ceived a name, dyne. It is the volume integral of force
density. The time integral of a field of force is momentum.
In a stationary field of force the line integral of the field
for a given path is the difference in energy between the
points at the ends of the path, or what is commonly called
work. In case the field is conservative the integral has the
same value for all paths (which at least avoid certain
singular points), and depends only on the end points,
This takes place when the field is a gradient field of a force-
function, or a potential function. If we project the force
upon the velocity at each point where both fields exist,
the time integral of the scalar quantity which is the product
of the intensity of the force, the intensity of the velocity
and the cosine of the angle between them, is the activity at
the point.

(8) Flux Density. 12 [UT~1}. In the case of the flow of
an entity through a surface the limiting value of the amount
that flows normally across an infinitesimal area is a vector
whose direction is that of the outward normal of the surface,
and whose intensity is the limit. In the case of a flow not
normal to the surface across which the flux is to be de-
termined, we nevertheless define the flux density as above.
The flux across any surface becomes then the surface
integral of the projection of the flux density on the normal
of the surface across which the flux is to be measured.

Flux density is an example of a vector which depends
upon an area, and is sometimes called a bivector. The
notion of two vectors involved in the term bivector may

30 VECTOR CALCULUS

be avoided by the term cycle, or the term feuille. It is
also called an axial vector, in opposition to the ordinary
vectors, called polar vectors. The term axial is applicable
in the sense that it is the axis or normal of a portion of a
surface. The portion (feuille, cycle) of the surface is
traversed in the positive direction in going around its
boundary, that is, with the surface on the left-hand. If
the direction of the axial vector is reversed, we also traverse
the area attached in the reverse direction, so that in this
sense the axial vector may be regarded as invariant for
such change while the polar vector would not be invariant.
The distinction is not of much importance. The important
idea is that of areal integration for the flux density or any
other so-called axial vector, while the polar vector is sub-
ject only to linear integration. We meet the distinction
in the difference below between the induction vectors and
the intensity vectors.

(9) Energy Density Current. R [TOL-2?7-2]. When an
energy density has the idea of velocity attached to it, it
becomes a vector with the given dimensions. In such
case we consider it as of the nature of a flux density.

(10) Energy Current. 2 [$QT~2]. If a vector of energy
density current is multiplied by an area we arrive at an
energy current.

(11) Electric Density Current. J [SL^T-1]. A number
of moving electrons will determine an average density
per square centimeter across the line of flow, and the product
of this into a velocity will give an electric density current.
To this must also be added the time rate of change of
electric induction, which is of the same dimensions, and
counts as an electric density current.

(12) Electric Current. C [971-1]. The unit is the ampere
= 3-109 e.s. units = 10_1 e.m. units. This is the product
of an electric density current by an area.

VECTOR FIELDS 31

(13) Magnetic Density Current. G [$Ir2T-1}. Though
there is usually no meaning to a moving mass of magnetism,
nevertheless, the time rate of change of magnetic induction
must be considered to be a current, similar to electric
current density.

(14) Magnetic Current. K [^T'1]. The unit is the
heavy side = 1 e.m. unit = 3 • 1010 e.s. units. In the phenom-
ena of magnetic leakage we have a real example of what may
be called magnetic current.

Both electric current and magnetic current may also be
scalars. For instance, if the corresponding flux densities
are integrated over a given surface the resulting scalar
values would give the rate at which the electricity or the
magnetism is passing through the surface per second. In
such case the symbols should be changed to corresponding
Roman capitals.

(15) Electric Intensity. E fMr1!1"1]. When an electric
charge is present in any portion of space, there is at each
point of space a vector of a field called the field of electric
intensity. The same situation happens when lines of
magnetic induction are moving through space with a given
velocity. The electric intensity will be perpendicular to
both the line of magnetic induction and to the velocity it
has, and equal to the product of their intensities by the
sine of their angle.

The electric intensity is of the nature of a polar vector
and its flux, or surface integral over any surface has no
meaning. Its line integral along any given path, however,
is called the difference of voltage between the two points at
the ends of the path, for that given path. The unit of
voltage is the volt = J • 10~2 e.s. units = 108 e.m. units.
The symbol for voltage is V [$T~1]. Its dimensions are
the same as for scalar electric potential, or magnetic current.

32 VECTOR CALCULUS

(16) Electric Induction. D [QL~2]. The unit is the line
= 3-109 e.s. units — 10-1 e.m. units. This vector usually
has the same direction as electric intensity, but in non-
isotropic media, such as crystals, the directions do not agree.
It is a linear function of the intensity, however, ordinarily

indicated by

D = k(E)

where k is the symbol for a linear operator which converts
vectors into vectors, called here the permittivity, [0^>-1Z_1 T],
measurable in farads per centimeter. In isotropic media
k is a mere numerical multiplier with the proper dimensions,
which are essential to the formulae, and should not be
neglected even when k = 1. The flux is measured in
coulombs.

(17) Magnetic Intensity. H [eL"1?7"1]. The field due to
the poles of permanent magnets, or to a direct current
traversing a wire, is a field of magnetic intensity. In case
we have moving lines of electric induction, there is a field of
magnetic intensity. It is of a polar character, and its
flux through a surface has no meaning. The line integral
between two points, however, is called the gilbertage between
the points along the given path, the unit being the gilbert
= 1 e.m. unit = 3 • 1010 e.s. units. The symbol is N [GT-1]'
Its dimensions are the same as those of scalar magnetic
potential, or electric current.

(18) Magnetic Induction. B [$L~2]. The unit is the
gauss = 1 e.m. unit = 3 • 1010 e.s. units. The direction is
usually the same as that of the intensity, but in any case is
given by a linear vector operator so that we have

B-m(H)

where \x is the inductivity, [^>0-1Z_1 T], measurable in henrys
per centimeter. The flux is measured in maxwells.

VPPf

VECTOR FIELDS 33

(19) Vector Potential of Electric Induction. T [eZ-1]. A
vector field may be related to another vector field in a
certain manner to be described later, such that the first
can be called the vector potential of the other.

(20) Vector Potential of Magnetic Induction. ^ [M-1].
This is derivable from a field of magnetic induction. This
and the preceding are line-integrable.

(21) Hertzian Vectors. 9, <£. These are line integrals of
the preceding two, and are of a vector nature.

5. Vector Lines. If we start at a given point of a vector
field and consider the vector of the field at that point to be
the tangent to a curve passing through the point, the field
will determine a set of curves called a congruence, since there
will be a two-fold infinity of curves, which will at every
point have the vector of the field as tangent. If the field
is represented by a, a function of p, the vector to a point of
the field, then the differential equation of these lines of
the congruence will be

dp = adt,

where dt is a differential parameter. From this we can
determine the equation of the lines of the congruence, in-
volving an arbitrary vector, which, however, will not have
more than two essential constants. For instance, if the
field is given by a = p, then dp = pdt, and p = ael, where
a is a constant unit vector. The lines are, in this case, the
rays emanating from the origin.

The lines can be constructed approximately by starting
at any given point, thence following the vector of the field
for a small distance, from the point so reached following
the new vector of the field a small distance, and so proceed-
ing as far as necessary. This will trace approximately a
vector line. Usually the curves are unique, for if the field
is monodromic at all points, or at points in general, the

34 VECTOR CALCULUS

curves must be uniquely determined as there will be at any
point but one direction to follow. Two vector lines may
evidently be tangent at some point, but in a monodromic
field they cannot intersect, except at points where the in-
tensity of the field is zero, for vectors of zero intensity are
of indeterminate direction. Such points of intersection
are singular points of the field, and their study is of high
importance, not only mathematically but for applications.
In the example above the origin is evidently a singular
point, for at the origin a = 0, and its direction is indetermi-
nate.

6. Vector Surfaces, Vector Tubes. In the vector field
we may select a set of points that lie upon a given curve
and from each point draw the vector line. All such vector
lines will lie upon a surface called a vector surface, which in
case the given curve is closed, forming a loop, is further
particularized as a vector tube. It is evident that the vector
lines are the characteristics of the differential equation
dp = adt, which in rectangular coordinates would be
equivalent to the equations

dx _dy _ dz
X ~ Y~ Z'

In case these equations are combined so as to give a
single exact equation, the integral will (since it must con-
tain a single arbitrary constant) be the equation of a family
of vector surfaces. The vector lines are the intersections
of two such families of vector surfaces. The two families
may be chosen of course in infinitely many different ways.
Usually, however, as in Meteorology, those surfaces are
chosen which have some significance. When a vector
tube becomes infinitesimal its limit is a vector line.

7. Isogons. If we locate the points at which a has the

VECTOR FIELDS 35

same direction, they determine a locus called an isogon for
the field. For instance, we might locate on a weather map
all the points which have the same direction of the wind.
If isogons are constructed in any way it becomes a simple
matter to draw the vector lines of the field. Machines for
the use of meteorologists intended to mark the isogons
have been invented and are in use.* As an instance con-
sider the vector field

a = (2x, 2y, — z).

An isogon with the points at which a has the direction whose
cosines are /, m, n is given by the equations

2x : 2y : — z = I : m : n
or

2x = It, 2y = mt, z = — nt.

It follows that the vector to any point of this isogon is
given by

p = t(l, m, n) - (0, 0, 3nt).

That is to say, to draw the vector p to any point of the
isogon we draw a ray from the origin in the direction given,
then from its outer end draw a parallel to the Z direction
backward three times the length of the Z projection of the
segment of the ray. The points so determined will evi-
dently lie on straight lines in the same plane as the ray and
its projection on the XY plane, with a negative slope twice
the positive slope of the ray. The tangents of the vector
lines passing through the points of the isogon will then be
parallel to the ray itself. The vector lines are drawn ap-
proximately by drawing short segments along the isogon
parallel to its corresponding ray, and selecting points such
that these short segments will make continuous lines in
*Sandstrdm: Annalen der Hydrographie und Maritimen Meteor-
ologie (1909), no. 6, pp. 242 et.seq. Bjerknes: Dynamic Meteorology.
See plates, p. 50.

36

VECTOR CALCULUS

passing to adjacent isogons. The figure illustrates the
method. All the vector lines are found by rotating the
figure about the X axis 180°, and then rotating the figure
so produced about the Z axis through all angles.

Fig. 3.

8. Singularities. It is evident in the example preceding
that there are in the figure two lines which are different
from the other vector lines, namely, the Z axis and the line
which is in the XY plane. Corresponding to the latter
would be an infinity of lines in the XY plane passing through
the origin. These lines are peculiar in that the other vector
lines are asymptotic to them, while they are themselves
vector lines of the field. A method of studying the vector
lines in the entire extent of the plane in which they lie was
used by Poincare. It consists in placing a sphere tangent

VECTOR FIELDS 37

to the plane at the origin. Lines are then drawn from the
center of the sphere to every point of the plane, thus giving
two points on the sphere, one on the hemisphere next the
plane and one diametrically opposite on the hemisphere
away from the plane. The points at infinity in the plane
correspond to the equator or great circle parallel to the
plane. In this representation every algebraic curve in the
plane gives a closed curve or cycle on the sphere. In the
present case, the axes in the plane give two perpendicular
great circles on the sphere, and the vector lines will be
loops tangent to these great circles at points where they
cross the equator. These loops will form in the four Junes
of the sphere a system of closed curves which Poincare calls
a topographical system. The equator evidently belongs to
the system, being the limit of the loops as they grow nar-
rower. The. two great circles corresponding to the axes
also belong to the system, being the limits of the loops as
they grow larger. If a point describes a vector line its
projection on the sphere will describe a loop, and could
never leave the lune in which the projection is situated.
The points of tangency are called nodes', the points which
represent the origin, and through which only the singular
vector lines pass, are called fames.

9. Singular Points. The simplest singular lines depend
upon the singular points and these are found comparatively
simply. The singular points occur where

o" = 0 or a —• oo .

Since we may multiply the components of a by any ex-
pressions and still have the lines of the field the same, we
may equally suppose that the components of a are reduced
to as low terms as possible by the exclusion of common
factors of all of them. We will consider first the singular

38 VECTOR CALCULUS

points for fields in space, then those cases which have
lines every point of which is a singular point, which will
include the cases of plane fields, since these latter may be
considered to represent the fields produced by moving the
plane field parallel to itself. The classification given by
Poincare is as follows.

(1) Node. At a node there may be many directions
in which vector lines leave the point. An example is a = p.
At the origin, it is easy to see, a = 0, and it is not possible
to start at the origin and follow any definite direction.
In fact the vector lines are evidently the rays from the
origin in all directions. There is no other singular point at
a finite distance. If, however, we consider all the rays in
any one plane, and for this plane construct the sphere of
projection, we see that the lines correspond to great circles
on the sphere which all pass through the origin and the
point diametrically opposite to it. This ideal point may
be considered to be another node, so that all the vector
lines run from node to node, in this case. Every vector
line which does not terminate in a node is a spiral or a cycle.

(2) Faux. From a faux* there runs an infinity of vector
lines which are all on one surface, and a single isolated
vector line which intersects the surface at the faux. The
surface is a singular surface since every vector line in it
through the faux is a singular line. The singular surface
is approached asymptotically by all the vector lines not
singular.

An example is given by

a = (x, y, — z).

The vector lines are to be found by drawing all equilateral
hyperbolas in the four quadrants of the ZX plane, and then
* Poincare uses the term col, meaning mountain pass, for which faux
is Latin.

/

VECTOR FIELDS

39

rotating this set of lines about the Z axis. Evidently all
rays in the XY plane from the origin are singular lines, as
well as the Z axis. Where fauces occur the singular lines
through them are asymptotes for the nonsingular lines. If

Fig. 4.

we consider any plane through the Z axis, the system of
equilateral hyperbolas will project onto its sphere as cycles
tangent on the equator to the great circles which repre-
sent the singular lines in that plane. From this point of
view we really should consider the two rays of the Z axis as
separate from each other, so that the upper part of the Z
axis and the singular ray perpendicular to it, running in the
same general direction as the other vector lines, would con-
stitute a vector line with a discontinuity of direction, or
with an angle. Such a vector line to which the others are
tangent at points at infinity only is a boundary line in the
sense that on one side we have infinitely many vector lines
which form cycles (in the sense defined) while on the other
sides we have vector lines which belong to different sys-
tems of cycles.

40

VECTOR CALCULUS

A simple case of this example might arise in the inward
flow of air over a level plane, with an ascending motion
which increased as the air approached a given vertical
line, becoming asymptotic to this vertical line. In fact,
a small fire in the center of a circular tent open at the bottom
for a small distance and at the vertex, would give a motion
to the smoke closely approximating to that described.

A singular line from a faux runs to a node or else is a
spiral or part of a cycle which returns to the faux.

An example that shows both preceding types is the field

a = (x2 + y2 — 1, bxy — 5, mz).

In the X Y plane the singular points are at infinity as follows :
A at the negative end of the X axis, and B at the positive
end, both fauces; C at the end of the ray whose direction
is tan-1 2, in the first quadrant, D at the end of the ray of
direction tan-1 2 in the third quadrant; E at the end of the

VECTOR FIELDS 41

ray of direction tan-1 — 2 in the fourth quadrant; and F
at the end of the ray of tan-1 — 2 in the second quadrant,
these four being nodes. Vector lines run from E to D
separated from the rest of the plane by an asymptotic
division line from B to D; from C to D on the other side
of this division line, separated from the third portion of
the plane by an asymptotic division line from C to A ; and
from C to F in the third portion of the plane. The figure
shows the typical lines of the field.

(3) Focus. At a focus the vector lines wind in asymp-
totically, either like spirals wound towards the vertex of a
spindle produced by rotating a curve about one of its
tangents, one vector line passing through the focus, or
they are like spirals wound around a cone towards the

Fig. 6.
vertex. As an example

o- = (x+ y, y - x, z).

The Z axis is a single singular line through the origin, which
is a singular point, a focus in this case. The XY plane
contains vector lines which are logarithmic spirals wound in
towards the origin. The other vector lines are spirals

42

VECTOR CALCULUS

wound on cones of revolution, their projections on XY
being the logarithmic spirals. By changing z to az we
would have different surfaces depending upon whether

1 < a.

a< 1

or

In case a spiral winds in onto a cycle, the successive
turns approaching the cycle asymptotically, the cycle is
called a limit cycle. In this example the line at infinity
in the X Y plane, or the corresponding equator on its sphere,
is a limit cycle. It is clear that the spirals on the cones
wind outward also towards the lines at infinity as limit
cycles. From this example it is plain that vector lines
which are spiral may start asymptotically from a focus and
be bounded by a limit cycle. The limit cycle thus divides
the plane or the surface upon which they lie into two
mutually exclusive regions. Vector lines may also start
from a limit cycle and proceed to another limit cycle.

As an example of vector lines of both kinds consider the
field

Fig. 7.
a = (r2 _ 1, r2 + lf mz)f

where the first component is in the direction of a ray in the
XY plane from the origin, the second perpendicular to

VECTOR FIELDS

43

this in the XY plane, and the third is parallel to the Z axis.
The vector lines in the singular plane, the XY plane, are
spirals with the origin as a focus for one set, which wind
around the focus negatively and have the unit circle as a
limit cycle, while another set wind around the unit circle
in the opposite direction, having the line at infinity as a
limit cycle. The polar equation of the first set is r~l — r

An example with all the preceding kinds of singularities
is the field

Fig. 8.

a = ( [r2 - l)(r - 9)], (r2 - 2r cos 9 - 8), mz)

with directions for the components as in the preceding
example. The singular points are the origin, a focus; the
point A (r = 3, 0 = + cos-1 §), a node; the point B (r = 3,
6 = — cos-1 J), a faux. The line at infinity is a limit
cycle, as well as the circle r = 1, which is also a vector
line. The circle r = 3 is a vector line which is a cycle,

4

44 VECTOR CALCULUS

starting at the faux, passing through the node and returning
to the faux. The vector lines are of three types, the first
being spirals that wind asymptotically around the focus,
out to the unit circle as limit cycle; the second start at the
node A and wind in on the unit circle as limit cycle; the
third start at the node A and wind out to the line at in-
finity as unit cycle. The second set dip down towards the
faux. The exceptional vector lines are the line at infinity,
the unit circle, both being limit cycles; the circle of radius
3; a vector line which on the one side starts at the faux B
winding in on the unit circle, and on the other side starts
at the faux B winding outward to the line at infinity as
limit cycle. The last two are asymptotic division lines of
the regions. The figure exhibits the typical curves.

(4) Faux-Focus. This type of singular point has passing
through it a singular surface which contains an infinity
of spirals having the point as focus, while an isolated vector
line passes through the point and the surface. No other
surfaces through the vector lines approach the point. An
instance is the field

a- = (x, y, — z).

The Z axis is the isolated singular line, while the XY plane
is the singular plane. In it there is an infinity of spirals
with the origin as focus and the line at infinity as limit
cycle. All other vector lines lie on the surfaces rz = const.
These do not approach the origin.

(5) Center. At a center there is a vector line passing
through the singular point, and not passing through this
singular line there is a singular surface, with a set of loops
or cycles surrounding the center, and shrinking upon it.
There is also a set of surfaces surrounding the isolated
singular line like a set of sheaths, on each of which there are
vector lines winding around helically on it with a decreasing

VECTOR FIELDS

45

Fig. 9.

pitch as they approach the singular surface, which they
therefore approach asymptotically. As an instance we
have the field

a = (y, - x, z).

The Z axis is the singular isolated vector line, the XY plane

the singular surface, circles

concentric to the origin the

singular vector lines in it, and

the other vector lines lie on

circular cylinders about the

Z axis, approaching the XY

plane asymptotically.

The method of determining
the character of a singular
point will be considered later
in connection with the study
of the linear vector operator.
A singular point at infinity is either a node or a faux.

10. Singular Lines. Singularities may not occur alone
but may be distributed on lines every point of which is a
singular point. This will evidently occur when cr = 0 gives
three surfaces which intersect in a single line. The dif-
ferent types may be arrived at by considering the line of
singularities to be straight, and the surfaces of the vector
lines with the points of the singular line as singularities
to be planes, -for the whole problem of the character of the
singularities is a problem of analysis situs, and the deforma-
tion will not change the character. The types are then as
follows :

(1) Line of Nodes. Every point of the singular line is a
node. A simple example is a = (x, y, 0). The vector
lines are all rays passing through the Z axis and parallel
to the XY plane.

46 VECTOR CALCULUS

(2) Line of Fauces. There are two singular vector
lines through each point of the singular line. As an instance
a = (x, — y, 0). The lines through the Z axis parallel to
the X and the Y axes are singular, all other vector lines
lying on hyperbolic cylinders.

(3) Line of Foci. The points of the singular line are
approached asymptotically by spirals. As an instance
<t = (x + y, y — x, 0). The vector lines are logarimithic
spirals in planes parallel to the XY plane, wound around the
Z axis which is the singular line.

(4) Line of Centers. A simple case is a — (y, — x, 0).
The vector lines are the Z axis and all circles with it as axis.

11. Singularities at Infinity. The character of these is
determined by transforming the components of a so as to
bring the regions at infinity into the finite parts of the
space we are considering. The asymptotic lines will then
have in the transformed space nodes at which the lines are
tangent to the asymptotic line.

12. General Characters. The problem of the character
of a vector field so far as it depends upon the vector lines
and their singularities is of great importance. Its general
resolution is due to Poincare. In a series of memoirs in
the Journal des Mathematiques* he investigated the
qualitative character of the curves which represent the
characteristics of differential equations, particularly with
the intention of bringing the entire set of integral curves
into view at once. Other studies of differential equations
usually relate to the character of the functions defined at
single points and in their vicinities. The chief difficulty
of the more general study is to ascertain the limit cycles.
These with the asymptotic division lines separate the
field into independent regions.

* Ser. (3) 7 (1881), p. 375; ser. (3) 8 (1882), p. 251; ser. (4) 1 (1885),
p. 167. Also Takeo Wado, Mem. Coll. Sci. Tokyo, 2 (1917) 151.

VECTOR FIELDS 47

The asymptotic division lines appear on meteorological
maps as lines on the surface of the earth towards which,
or away from which, the air is moving. They are called
in the two cases lines of convergence, or lines of divergence,
respectively. If a division line of this type starts at a
node the node may be a point of convergence or a point of
divergence. The line will then have the same character.
The node in other fields, such as electric or magnetic or heat
flow, is a source or a sink. If a division line starts from a
faux, the latter is often called a neutral point. A focus may
be also a point of convergence or point of divergence. In
the case of a singular line consisting of foci, the singular line
may be a line of convergence or of divergence; in the first
case, for instance, the singular line is the core of the anti-
cyclone, in the latter case, the core of the cyclone.

The limit cycles which are not at infinity are division
lines which enclose areas that remain isolated in the field.
Such phenomena as the eye of the cyclone illustrate the oc-
currence of limit cycles in natural phenomena. The limit
cycle may be a line of convergence or a line of divergence,
the air in the first case flowing into the line asymptotically
from both inside and outside, with the focus serving as a
source, and in the other case with conditions reversed.

The practical handling of these problems in meteorological
work depends usually upon the isogonal lines: the lines
which are loci of equidirected tangents of the vector lines
of the field. These are drawn and the infinitesimal tan-
gents drawn across them. The filling in of the vector
lines is then a matter of draughtsmanship. The isogonal
lines will themselves have singularities and these will
enable one to determine somewhat the singularities of the
vector lines themselves. Since the unit vector in the
direction of a is constant along an isogon it is evident that

48 VECTOR CALCULUS

the only change in a along an isogon is in its intensity,
that is, a keeps the same direction, and its differential is
therefore a multiple of a, that is, the isogons have for their
differential equation

da = adt.

Consequently, when a = 0 or a = <x> the isogon will have a
singular point. It does not follow, however, that all the
singular points of the isogons will appear as singular points
such as are described above for the vector lines. When
the differential equation of the isogons is reduced to the
standard form

dp = rdu

we shall see later that r will be a linear vector function of a,
and that a linear vector function may have zero directions,
so that <pa — 0, without a = 0. Some of the phenomena
that may happen are the following, from Bjerknes' Dynamic
Meteorology and Hydrography. See his plates 42a, 426.

1. Node of Isogons. These may be positive, in which
case the directions of the tangents of the vector lines will
increase (that is, the tangent will turn positively) as succes-
sive isogons are taken in a positive rotation about the node,
or may be negative in the reverse case. The positive node
of the isogon will then correspond to a node, a focus, or a
center of the vector lines. The negative node of the isogon
will correspond to a faux of the vector lines.

If the isogons are parallel, having, therefore, a node at
infinity in either of their directions, the vector lines may
have asymptotic division lines running in the same direc-
tion, or they may have lines of inflexion parallel to the
isogons.

2. Center of Isogons. When the isogons are cycles they
may correspond to very complicated forms of the vector

VECTOR FIELDS 40

lines. Several of these are to be found in a paper by Sand-

strom, Annalen der Hydrographie und maritimen Meteor-

ologie, vol. 37 (1909), p. 242, Uber die Bewegung der

Flussigkeiten.

EXERCISES*

* To be solved graphically as far as possible.

1. A translation field is given by a- = (at, bt, ct), what are the vector
lines, the isogons, and the singularities?

2. A rotation field is given by a = (mz — ny, nx — Iz, ly — mx),
what are the isogons, singularities, and vector lines?

3. A field of deformation proportional to the distance in one direction
is given by a = {ax, 0, 0). Determine the field.

4. A general field of linear deformation is given by

o- = (ax + by + cz, fx + gy + hz, kx + ly + tnz) .
determine the various kinds of fields this may represent according to
the different possible cases.

5. Consider the quadratic field*

a = (x2 — y2 — z2, 2xy, 2xz).

6. Consider the quadratic field a = (xy — xz, yz — yx, zx — zy).

7. What are the lines of flow when the motion is stationary in a
rotating fluid contained in a cylindrical vessel with vertical axis of
rotation?

8. Consider the various fields a = (ay -\- x, y — ax, b) for different
values of a, which is the tangent of the angle between the curves and
their polar radii. What happens in the successive diagrams to the
isogons, to the curves?

9. Consider the various fieldsf a = (l,f(r — a), b) where r is the
polar radius in the XY plane, a is constant, and / takes the various
forms

f(x) = x, x2, x3, x112, x113, x~l, x~2, ex, log x, sin x, tan x.

10. Consider the forms a = (1, f(air sin r), b) where

j(x) = sin x, cos x, tan x.

11. In various electrical texts, such as Maxwell, Electricity and
Magnetism, and others, there will be found plates showing the lines of
various fields. Discuss these. Also, the meteorological maps in
Bjerknes' Dynamic Meteorology, referred to fibove.

* See Hitchcock, Proc. Amer. Acad. Arts and Sci., 12 (1917), No. 7,
pp. 372-454.

f See Sandstrom cited above.

50 VECTOR CALCULUS

12. In a funnel-shaped, vortex of a water-spout the spout may be
considered to be made up of twisted funnels, one inside another, the
space between the surfaces being a vortex tube. In the Cottage City
water-spout, Aug. 19, 1896, the equation of the outside funnel may be
taken to be

(z2 + y*)z = 3600.

In this x, y are measured horizontally in meters from the axis of the
tubes, and z is measured vertically downwards from the cloud base,
which is 1100 meters above the ground. The inner surfaces have the
same equation save that instead of 3600 on the right we have
3600/(1. 60 10)2n; that is, at any level, the radius of a surface bounding
a tube is found from the preceding radius at the same level by dividing
by the number whose logarithm (base 10) is 0.20546. From meteoro-
logical theory the velocity of the wind on any surface is given by

<r = (Cr, Crz, - 2Cz)

where the first component is the horizontal radial component, the
second is the tangential, and the third is the vertical component. C
varies for the different surfaces, and is found by multiplying the value
for the outside surface by the square of the number 1.6010. In Bige-
low's Atmospheric Radiation, etc., p. 200 et seq., is to be found a set of
tables for the various values from these data for different levels. Char-
acterize the vortex field of the water-spout.

13. For a dumb-bell-shaped water-spout, likewise, the funnels have

the equation

(x2 + y2) sin az — const/A

where A varies from surface to surface just as C in the preceding
problem. The velocity is given by

o- = (— Aar cos az, Aar sin az, 2A sin az),

the directions being horizontal radial, tangential and vertical. For
the St. Louis tornado, May 27, 1896, the following data are given.
Cloud base 1200 meters above the ground, divided into 121 parts
called degrees, the ground thus being at 60°, and az being in degrees.
The values of A are for the successive funnels

0.1573, 0.4052, 1.0437, 2.6883, 6.9247, 17.837.

Characterize the vector Ikies of this vortex field.

14. In the treatise on The Sun's Radiation, Bigelow gives the follow-
ing data for a funnel-shaped vortex

r2z = 6400000/C

W=wind--from9f

PLATE I

PLATE II

VECTOR FIELDS 51

at 500 kilometers z = 500, r = 60474, 26287, 11513, 5023, 2192, 956.

a (Km/sec) = (Cr, Crz, - 2Cz).
Calculate for

z = 0, 500, 1000, 2000, 5000, 10000, 20000, 30000, 40000, 50000.

The results of the calculations give a vortex field agreeing with Hale's
observations.

The vector lines in the last three problems lie on the funnel surfaces,
being traced out in fact by a radius rotating about the axis of the vortex,
and advancing along the axis according to the law

2d = - z + C for the funnel,
20 = az + C for the dumb-bell.

15. Study the lines on the plates, which represent on the first plate
the isogons for wind velocities, on the second plate the corresponding
characteristic lines of wind flow. The date was evening of Jan. 9, 1908.
European and American systems of numbering directions are shown in
the margin of plate 1. See Sandstrom's paper cited above.

13. Congruences. We still have to consider the relations
of the various vector lines to each other, noticing that the
vector lines constitute geometrically a congruence, that is,
a two-parameter system of curves in space. The con-
sideration of these matters, however, will have to be post-
poned to a later chapter.

CHAPTER IV

ADDITION OF VECTORS

1 . Sum of Vectors. Geometrically, the sum of two or
more vectors is found by choosing any one of them as the
first, from the terminal point of the first constructing the
second (any other), from the terminal point of this con-
structing the third (any of those left) and so proceeding
till all have been successively joined to form a polygon in
space with the exception of a final side. If now this last
side is constructed by drawing a vector from the initial
point of the first to the terminal point of the last, the vector
so drawn is called the sum of the several vectors. In
case the polygon is already closed the sum is a zero vector.
When the sum of two vectors is zero they are said to be
opposite, and subtraction of a vector consists in adding its
opposite.

It is evident from the definition that we presuppose a space in which
the operations can be effectively carried out. For instance, if the space
were curved like a sphere, and the sum of two vectors is found, it would
evidently be different according to which is chosen as the first. The
study of vector addition in such higher spaces has, however, been con-
sidered. Encyclopedic des sciences mathematiques, Tome IV, Vol. 2.

2. Algebraic Sum. In order to define the sum without
reference to space, it is necessary to consider the hyper-
numbers that are the algebraic representatives of the
geometric vectors. We must indeed start with a given
set of hypernumbers,

which are the basis of the system of hypernumbers we in-
tend to study. These are sometimes called imaginaries,
because they are analogous to V— 1. In the case of three-

52

ADDITION OF VECTORS 53

dimensional space there are three such hypernumbers in the
basis. We combine in thought a numerical value with
each of these, the field or domain from which these numeri-
cal values are chosen being of great importance. For in-
stance, we may limit our numbers to the domain of integers,
the domain of rationals, the domain of reals, or to other
more complicated domains, such as certain algebraic fields.
We then consider all the multiplexes we can form by put-
ting together into a single entity several of the hypernum-
bers just formed, as for instance, we would have in three-
dimensional space such a compound as

p = («1, 7/e2, Z€3).

Since we are now using the base hypernumbers e it is no
longer necessary to use the parentheses nor to pay attention
to the order of the terms. We drop the use of the comma,
however, and substitute the + sign, so that we would now
write

p = X€i + 2/€2 + 2€3.

We may now easily define the algebraic sum of several
hypernumbers corresponding to vectors by the formula

Pi = Xi€i + y{€2 + Zi*z,[ i = 1, 2, • • • m,

]T Pi = 2£i€i + 2^€2 + 2Zi€3.

i = 1

This definition of course includes subtraction as a special
case.

It is clear from this definition that to correspond to the
geometric definition, it is necessary that the units e corre-
spond to three chosen unit vectors of the space under con-
sideration. They need not be orthogonal, however. The
coefficients of the e are then the oblique or rectangular
coordinates of the point which terminates the vector if it
starts at the origin.

54 VECTOR CALCULUS

3. Change of Basis. We may define all the hyper-
numbers of the system in terms of a new set linearly related
to the original set. For instance, if we write

€1 = duOti + ai2«2 + Ol3«3,
€2 = CiziOLl + a22«2 "T" 023«3>

€3 = a3iai + a32a2 + «33«3,
then p becomes

P = (anx + any + anz)ai

+ (aux + a22y + a32z)a2 + (anx -f a2zy + a33z)a3.

It is evident then that if we transform the e's by a non-
singular linear homogeneous transformation, the coeffi-
cients of the new basis hypernumbers, a, are the transforms
of the original coefficients under the contragredient trans-
formation.

Inasmuch as the transformation is linear, the transform of
a sum will be the sum of the transforms of the terms of the
original sum. The transformation as a geometrical process
is equivalent to changing the axes. This process evidently
gives us a new triple, but must be considered not to give
us a new hypernumber nor a new vector. Indeed, a vector
cannot be defined by a triple of numbers alone. There
is also either explicitly stated or else implicitly understood
to be a basis, or on the geometric side a definite set of axes
such that the triple gives the components of the vector
along these axes. It is evident that the success of any
system of vector calculus must then depend upon the
choice of modes of combination which are not affected by
the change from one basis to another. This is the case
with addition as we have defined it. We assume that we
may express any vector or hypernumber in terms of any
basis we like, and usually the basis will not appear.

If the transformation is such as to leave the angles be-

ADDITION OF VECTORS 55

tween ei, e2, e3 the same as those between a\, a2, a3, the
second trihedral being substantially the same as the first
rotated into a new position, with the lengths in each case
remaining units, then the transformation is called orthog-
onal. We may define an orthogonal transformation algebra-
ically as one such that if followed by the contragredient
transformation the original basis is restored.

4. Differential of a Vector. If we consider two points
at a small distance apart, the vector to one being p, to the
other p', and the vector from the first to the second, Ap
= p' — p, where Ap = As-e, e being a unit vector in the
direction of the difference, we may then let one point ap-
proach the other so that in the limit e takes a definite posi-
tion, say a, and we may write ds for As, and call the result
the differential of p for the given range over which the pf
runs. In the hypernumbers we likewise arrive at a hyper-
number

dp = dxei -f- dye?, + dzez,

where now ds is a linear homogeneous irrational function
of dx, dy, dz, which = V (dx2 + dy2 + dz2) in case ely e2, e3
form a trirectangular system of units.

The quotient dpjdt is the velocity at the point if t repre-
sents the time. The unit vector a: is the unit tangent for
a curve. We generally represent the principal normal and
the binormal by jS, 7 respectively. When p is given as
dependent on a single variable parameter, as t for instance,
then the ends of p may describe a curve. We may have
in the algebraic form the coordinates of p alone dependent
upon the parameter, or we may have both the coordinates
and the basis dependent upon t. For instance, we may ex-
press p in terms of ei, e2, e3 which are not dependent upon
t but represent fixed directions geometrically, or we may
express p in terms of three hypernumbers as w, r, J* which

56 VECTOR CALCULUS

themselves vary with t, such as the moving axes of a system
in space. In relativity theories the latter method of repre-
sentation plays an important part.

5. Integral of a Vector. If we add together n vectors and
divide the result by n we have the mean of the n vectors,
which may be denoted by p. If we select an infinite
number of vectors and find the limit of their sum after
multiplication by dt, the differential of the parameter by
which they are expressed, such limit is called the integral
of the vector expressed in terms of t, and if we give t two
definite values in the integral and subtract one result from
the other, the difference is the integral of the vector from
the first value of t to the second. More generally, if we
multiply a series of vectors, infinite in number, by a corre-
sponding series of differentials, and find the limit of the
sum of the results, such limit, when it exists, is called the
integral of the series. In integration, as in differentiation,
the usual difficulties met in analysis may appear, but as
they are properly difficulties due to the numerical system
and not to the hypernumbers, we will suppose that the
reader is familiar with the methods of handling them.

The mean in the case of a vector which has an infinite

sequence of values is the quotient of the integral taken on

some set of differentials, divided by the integral of the set

of differentials itself. The examples will illustrate the

use of the mean.

EXAMPLES

(1) The centroid of an arc, an area, or a volume is found
by integrating the vector p itself multiplied by the dif-
ferential of the arc, ds, or of the surface, du, or of the volume
dv. The integral is then divided by the length of the
arc, the area of the surface, or the volume. That is

- Sheets ffpdu • fffpdv m

P — — , or - — or —

b— a A V

ADDITION OF VECTORS

57

(2) An example of average velocity \s found in the following
(Bjerknes, Dynamic Meteorology, Part II, page 14) obser-
vations of a small balloon.

2 = Ht. in

Meters

Az

Direction

Velocity

(w/sec.)

Products

77
680
960
1240
1530
1810
2090
2430
2730
3040
3400
3710
4030
4400

603
280
280
290
280
280
340
300
310
360
310
320
370

S. 50° E.
S. 57° E.
S. 36° E.
S. 28° W.
S. 2°W.
S. 2°W.
S. 35° W.
S. 53° W.
S. 69° W.
S. 55° W.
S. 53° W.
S. 58° W.
S. 37° W.

3.4
4.0
5.3
1.5
1.8
2.0
1.5
1.8
1.8
3.0
2.8
4.4
10.2

2050

1120

1484

435

504

560

510

540

558

1080

868

1408

3773

To average the velocities we notice that on the assump-
tion that the upward velocity was uniform the distances
vertically can be used to measure the time. We therefore
multiply each velocity by the difference of elevations
corresponding, the products being set in the last column.
These numbers are then taken as the lengths of the vectors
whose directions are given by the third column. The
sum of these is found graphically, and divided by the total
difference of distance upward, that is, 4323. In the same
manner we can find graphically the averages for each 1000
meters of ascent. We may now make a new table in order
to find other important data, as follows :

Height

. Pressure
(ra-bars)

Dens, (ton/w3)

Veloc. .

Spec. Mo-
mentum
(ton/ra2 sec.)

4000

3000

2000

1000

75

622
705
797
899
1003

0.00083
0.00092
0.00102
0.00112

3.8
1.6

• 2.4
3.7

0.0032
0.0015
0.0025
0.0041

;,s

VECTOR CALCULUS

We now find the average velocity between the 1000 m-bar,
the 900 ra-bar, the 800 m-bar, the 700 ra-bar, and the 600
ra-bar. The direction is commonly indicated by the in-
tegers from 0 to 63 inclusive, the entire circle being divided
into 64 parts, each of 5f°. East is 0, North is 16, NW. is
24, etc. The following table is found.

Pressure

Height

Spec. Vol.

(m3/Ton)

Direction

Veloc.

Spec. Mo-
mentum

600

700

800

' 900

1000

1002.6

4274

3057

1970

989

99

76

1217

1087

981

890

890

8

7

20

25

25

5.2
1.7
2.4
3.7
3.4

0.0043
0.0016
0.0024
0.0042
0.0040

Of course, specific momenta should be averaged like veloc-
ities but usually owing to the rough measurements it is
sufficient to find specific momenta from the average
velocities.

ADDITION OF VECTORS

59

EXERCISES

1. Average as above the following observations taken at places
mentioned (Bjerknes, p. 20), July 25, 1907, at 7 a.m. Greenwich time.

Isobar

Dyn. Ht.

Az

Direction

Veloc.

100

200

300

400

500

600

700

800

900
1000
1001.2

16374

11947

9320

7301

5648

4240

3020

1938

975

107

98

4427

2627

2019

1653

1408

1220

1082

963

867

9

0

10

18

19

8

5

4

4

36

35

4.7
3.2
3.4
3.3
2.6
2.5
2.5
1.4
4.5
4.5

Uccle,
Lat. 50° 48'
Long. 4° 22'

100
200
300
400
500
600
700
800
900
955.9

16238

11817

9240

7248

5626

4244

3038

1955

977

471

4421
2577
1992
1622
1382
1206
1083
978
506

3
6
7
2
3
2

62
4

30

10.0
6.5
7.6

10.2
6.7
6.8
5.3
0.6
2.1

Zurich,
Lat. 47° 23'
Long. 8° 33'

200
300
400
500
600
700
800
900
1000

11890

9241

7240

5643

4196

2991

1927

981

118

17

2649

2001

1597

1447

1205

1064

946

863

101

59
57
58
55
49
41
38
56
55

9.2
10.5
8.8
8.0
2.9
2.9
1.9
4.3
3.4

Hamburg,
Lat. 53° 33'
Long. 9° 59'

GO

VECTOR CALCULUS

2. If the direction of the wind is registered every hour how is the
average direction found? Find the average for the following observa-
tions.

Station

Pikes
Peak

Vienna

Mauritius

Cordoba

S Orkneys

Elev

4308 m.

26 m.

15 m.

437 m.

25 m

Summer

Winter

Dec-Feb.

Winter

Summer

Time

Vel.

Az.

Vel.

Az.

Vel.

Az.

Vel.

Az.

Vel.

Az.

0 a.m

0.84
1.34
1.46
1.05
0.43
0.66
1.03

100°
83
71
57
12
279
262

0.47
0.56
0.42
0.33
0.22
0.17
0.36

62
61
59
57
46
303
257

1.00
1.30
1.30
1.00
1.10
1.80
2.40

100
97
98
119
241
312
326

0.94
1.06
1.44
2.03
2.17
0.50
2.78

115
111
121
132
136
252
314

0.52
0.52
0.51
0.51
0.52
0.53
0.54

70

2

51

4

30

6

5

8

343

10

?m

12 noon

255

14

1.09

256

0.58

242

1.60

332

3.56

315

0.54

245

16

0.95

253

0.64

232

1.30

304

3.36

305

0.54

42

18

0.74

247

0.47

223

0.20

10

1.75

299

0.53

35

20

0.49

47

0.14

186

0.90

101

0.72

44

0.53

50

22

0.36

153

0.25

72

1.00

102

0.89

128

0.52

60

Bigelow, Atmospheric Circulation, etc., pp. 313-315.

3. The following table gives the mean magnetic deflecting vectors,
in four zones, the intensity measured in 10~6 dynes, <p measured from
S. to E., N., W., and 0 is measured above the horizon. The vector is
the deflection from the mean position. Find the average for each
zone. (Bigelow, pp. 324-325.)

Time

Arctic

N Temperate

Tropic

S Temperate

s

e

<P

s

0

<P

s

0

<p

s

0

<p

a.m.

0

60

-36°

345°

15

-30°

111°

20

-33°

5°

19

27°

259°

1

63

-44

355

14

-35

109

19

-32

16

19

31

250

2

69

-43

5

14

-32

102

20

-36

7

17

35

251

3

74

-44

16

14

-33

108

20

-42

6

18

36

243

4

75

-42

25

15

-35

112

18

-34

10

20

36

226

5

77

-42

30

17

-33

110

17

-37

6

21

33

223

6

78

-40

32

20

-31

112

19

-36

4

24

31

222

7

76

-40

36

22

- 6

107

21

-37

339

26

24

235

8

65

-37

45

25

3

99

24

-30

297

28

28

248

9

54

-18

68

26

24

66

26

23

228

28

33

256

10

39

31

117

27

37

49

35

25

210

26

-27

296

11

47

44

195

25

38

312

43

22

204

25

-37

327

ADDITION OF VECTORS 61

4. Find the resultant attraction at a point due to a segment of a
straight line which is (a) of uniform density, (6) of density which varies
as the square of the distance from one end. What is the mean attrac-
tion in each case?

5. Show that p = ta + \P$ is the equation of a parabola, that the
equation of the tangent is p = Ua + \t\2& + x(a + ttfi), that tangents
from a given point are given by t = p ± V (p2 — 2q), the point being
pa + q/3, the chord of contact is p = — qP + y(a + PP) which has a
direction independent of q so that all points of the line p = pa + zP
have corresponding chords of contact which are parallel. If a chord
is to pass through the point aa + bp for differing values of p, then
q = ap — b and the moving point pa + qP lies on the line p — pa
-f- (ap — b)P, whose direction is independent of b.

6. If a, /S, 7 are vectors to three collinear points, then we can find
three numbers a, b, c such that

aa + 6/S + cy = 0 = a + b + c.

7. In problem 5 show that if three points are taken on the parabola
corresponding to the values t\, U, tz, then the three points of intersection
of the sides of the triangle they determine with the tangents at the
vertices of the triangle are collinear.

8. Determine the points that divide the segment joining A and B,
points with vectors a and 0, in the ratio I : m, both internally and ex-
ternally. Apply the result to find the polar of a point with respect to
a given triangle, that is, the line which passes through the three points
that are harmonic on' the three sides respectively with the intersection
of a line through the given point and the vertex opposite the side.

9. Show how to find the resultant field due to superimposed fields.

10. A curve on a surface is given by p = u(u, v), u = /(v), study the
differential of p.

CHAPTER V

VECTORS IN A PLANE
1. Ratio of Two Vectors. We purpose in this chapter to
make a more detailed study of vectors in a plane and the
hypernumbers corresponding. In the plane it is convenient
to take some assigned unit vector as a reference for all
others in the plane, though this is not at all necessary in
most problems. In fact we go back for a moment to the
fundamental idea underlying the metric notion of number.
According to this a number is defined to be the ratio be-
tween two quantities of the same concrete kind, such as
the ratio of a rod to a foot. If now we consider the ratio
of vectors, regarding them as the same kind of quantity,
it is clear that the ratio will involve more than merely
numerical ratio of lengths. The ratio in this case is in
fact what we have called a hypernumber. For every pair
of vectors p, x there exists a ratio p : x and a reciprocal
ratio x : p. This ratio we will designate by a roman
character

P

p : x = p/x =

IT

That is to say, we may substitute p for qw.

2. Complex Numbers. If we draw p and x from one
point, they will form a figure which has two segments for
sides and an angle. (In case they coincide we still con-
sider they have an angle, namely zero.) In this figure p is
the initial side and x is the terminal side. Then their
complex ratio is x : p. Since this ratio is to be looked upon
as a multiplier, it is clear that if we were to reduce the
sides in the same proportion, the ratio would not be changed.

62

VECTORS IN A PLANE 63

A change of angle would, however, give a different ratio.
However, we will agree that all ratios are to be considered
as equivalent, or as we shall usually say, equal, not only
when the figures to which they correspond have sides in
the same proportion, but also when they have the same
angles and sides in proportion, even if not placed in the
plane in the same position. For instance, if the vectors
AB, AC make a triangle which is similar to the triangle
DE, DF, if we take the sides in this order, then we shall
consider that whatever complex or hypernumber multiplies
AC into AB will also multiply DF into DE. This axiom
of equivalence is not only important but it differentiates
this particular hypernumber from others which might just
as well be taken as fundamental. For instance, the Gibbs
dyad of t : p is equally a hypernumber, but we cannot
substitute for ir or p any other vectors than mere multiples
of 7r or p. It is clear that in the Gibbs dyad we have a
more restricted hypernumber than in the ordinary com-
plex number which has been just defined, and which is a
special case of the Hamiltonian quaternion. If we have
a Gibbs dyad q, we can find the two vectors ir and p save
as to their actual lengths. But with the complex number
q we cannot find ir and p further than to say that for every
vector there is another in the ratio q. In other words the
only transformations allowed in the Gibbs dyad are transla-
tion of the figure AB, AC or magnification of it. In the
Hamiltonian quaternion, or complex number, the trans-
formations of the figure AB, AC may be not only those
just mentioned but rotation in the plane.

In order to find a satisfactory form for the hypernumber
q which we have characterized, we further notice that if
we change the length of x in the ratio m then we must
change q in the same ratio, and if we set for the ratio of the

64 VECTOR CALCULUS

length or intensity of w to that of p the number r, it is evi-
dent that we ought to take for q an expression of the form

q = r<p(0),

where <p(6) is a function of 0, the angle between p and t,
only. Further if we notice that we now have

7T = r(p(6)p,

where the first factor affects the change of length, the
second the change of direction, it is plain that for a second
multiplication by another complex number q' = r'<p(0'),
we should have

tt' = r,rcp(e,)iP{e)P = r'rip(W + 6)p.

Whence we must consider that

viO'Md) = <p{ef +$) = view).

These expressions are functions of two ordinary numerical
parameters, 0, 0', and are subject to partial differentiation,
just like any other expressions. Differentiating first as to
0, then as to 6', we find (<pf being the derivative)

<p\eM6') = ?'($+ ef) = wmb),

whence

. vy) = V'{6') _

where & is a constant and does not depend upon the angle at
all. It may, however, depend upon the plane in which
the vectors lie, so that for different planes A; may be, and
in fact is, different. N

Since, when 0 = 0 the hypernumber becomes a mere
numerical multiplier,

<p'(0) = MO).

If now we examine the particular function
<p(0) = cos 0+k sin 6,

VECTORS IN A PLANE 65

which gives

<p'(d) = — * sin $ + k cos 6 = k cos 6 + k2 sin 6,

we find all conditions are satisfied if we take k2 = — 1.
We may then properly use this function to define <p.
This very simple condition then enables us to define hyper-
numbers of this kind, so that we write

q = r(cos 6 + k sin 9) = r cks 6 = rg,

where k2 = — 1.

3. Imaginaries. It is desirable to notice carefully here
that we must take k2 equal to — 1, the same negative
number that we have always been using. This is important
because there are other points of view from which the
character of k and k2 would be differently regarded. For
instance, in the original paper of Hamilton, On Algebraic
Couples, the k, or its equivalent, is regarded as a linear
substitution or operator, which converts the couple (a, b)
into the couple (— b, a). While it is true that we may so
regard the imaginary, it becomes at once obvious that we
must then draw distinctions between 1 as an operator, and
1 as a number, and so for — 1, and indeed for any expression
x + yi. In fact, such distinctions are drawn, and we find
these operators occasionally called matrix unity, etc. From
the point of view of the hypernumber, this distinction is
not possible. Hypernumbers are extensions of the number
system, similar to radicals and other algebraic numbers.
The fact that, as we will see later, they are not in general
commutative, does not prevent their being an extension.

4. Real, Imaginary, Tensor, Versor. In the complex
number

q = r cos 6 + r sin 6 • k

the term r cos 6 is called the real part of q and may be written
Rq. The term r sin 6-k is called the imaginary part of q

66 VECTOR CALCULUS

and written Iq. The number r is called the tensor of q and
written Tq. The expression cos 6 + sin 0 • k is called the
versor of </ and written Uq. Therefore,

q= Rq+ Iq= TqUq.

If q appears in the form q — a + bk we see at once that

Rq = a, Iq= bk, Tq = V (a2 + b2), 6 = taiT^/a.

5. Division. If we have w = qp, then we also write
p = g-17r. It becomes evident that

&Tl = RqKTqf, Iq-' = - Iql(Tq)\ Tq-* = 1/Tq,
Uq-1 = cos 6 — sin 6 • k.

6. Conjugate, Norm. The hypernumber q = Kq — Rq
— Iq is called the conjugate of q. If q belongs to the figure
AB, AC, then q belongs to an inversely similar triangle, that
is, a similar triangle which has been reflected in some
straight line of the plane. The product q° = Nq = (Tq)2
is called the norm of q. It also has the name modulus of q,
particularly in the theory of functions of complex variables.

Evidently,

Rq = i(q + q), Iq = h(q - ~q), r1 = W* ^q~l = Uq-

7. Products of Complex Numbers. From the definitions
it is clear that the product of two complex numbers q, r,
is a complex number s, such that

Ts = TqTr,_ ZJ= zq+ Zr,
Rqr = Rrq = Rqr = Rrq = RqRr - Tlqlr,
Rqr = Rqr = Rrq = Rrq = RqRr + Tlqlr,
Iqr = Irq = — Tqr = — Irq = Rqlr + Rrlq,
Iqr = Irq = — Iqr = — Irq = Rrlq — Rqlr.

Hence if Rqr = 0, the angles of q and r are complementary
or have 270° for their sum.

VECTORS IN A PLANE 67

If Rqr = 0, the angles differ by 90°. In particular

we may take r = 1.
If Iqr = 0, the angles are supplementary or opposite.
If Iqr = 0, the angles are equal or differ by 180°.

8. Continued Products. We need only to notice that

(qrs- • -z) = (z- • -srq).

It is not really necessary to reverse the order here as the
products are commutative, but in quaternions, of which
these numbers are particular cases, the products are not
usually commutative, and the order must be as here
written.

9. Triangles. If ft y, 5, e are vectors in the plane, and

e = gft 5 = gyf

then the triangle of ft e is similar to that of y, 5, while if
e = gft 5 = ?7,

the triangles are inversely similar.

These equations enable us to apply complex numbers to
certain classes of problems with great success.

10. Use of Complex Numbers as Vectors. If a vector a
is taken as unit, every vector in the plane may be written
in the form qa, for some properly chosen q. We may
therefore dispense with the writing of the a, and talk of
the vector q, always with the implied reference to a certain
unit a. This is the well-known method of Wessel, Argand,
Gauss, and others. However, it should be noticed that
we have no occasion to talk of q as a point in the plane.

EXAMPLES
(1) Calculate the path of the steam in a two-wheel tur-
bine from the following data. The two wheels are rigidly
connected and rotate with a speed a = 4000° ft./sec. Be-

68 VECTOR CALCULUS

tween them are stationary buckets which turn the exhaust
steam from the buckets of the first wheel into those of the
second wheel. The friction in each bucket reduces the speed
by 12%. The steam issues from the expansion nozzle at a
speed of /3 = 22002o°. The proper exhaust angles of the
buckets are 24°, 30°, 45°. Find the proper entrance angles
of the buckets.

7 = relative velocity of steam at entrance to first wheel.
= 220020 - 400o = 183024.3.

8 = velocity of issuing steam, 88% of preceding,

= 1610x56.

€ = entrance velocity to stationary bucket.

= 5 + a = I6IO1M + 400o = 1255i48.4.
f = exit - 110530.
0 = entrance to next bucket = £ — a = 110530 — 400o

= 78044.3.
77 = exit = 69O135. Absolute exit velocity = 690i35
+ 4000 = 495ioo.

Steinmetz, Engineering Mathematics.

(2). We may suppose the student is somewhat familiar
with the usual elementary theory of the functions of a
complex variable. If w is an analytic function of z, both
complex numbers, then the real part of w, Rw, considered
as a function of x, y or u, v, the two parameters which de-
termine z, will give a system of curves in the x, y, or the
u, v plane. These may be considered to be the transforma-
tions of the curves Rw = const, which are straight lines
parallel to the Y axis in the w plane. Similarly for the
imaginary part. The two sets will be orthogonal to each

other, since the slope of the first set will be ^— / -z — ;

J * 1 1 dTIw/dTIw _

and 01 the other set ^ — / —^ — . But these are

ox I dy

VECTORS IN A PLANE 69

negative reciprocal, since

dRw dTIw dRw_ dTIw

~ — n and ~ — ~

ox oy ay ox

EXERCISES

1 . If a particle is moving with the velocity 12028° and enters a medium
which has a velocity given by

<r = P + 36 sin z [p, 0]8°,
what will be its path?

2. The wind is blowing steadily from the northwest at a rate of
16 ft. /sec. A boat is carried round in circles with a velocity 12 ft. /sec.
divided by the distance from the center. The two velocities are com-
pounded, find the motion of the boat if it starts at the point p = 40°.

3. A slow stream flows in at the point 120° and out at the point
12i8o°, the lines of flow being circles and the speed constant. A chip
is floating on the stream and is blown by the wind with a velocity
6400. Find its path.

4. If a triangle is made with the sides q, r then R.qr is the power of
the vertex with reference to the circle whose diameter is the opposite
side. The area of the triangle is \TIqr.

5. The sum q + r can be found by drawing vectors qa, ra.

6. How is qr constructed? qr?

7. If OAE is a straight line and OCF another, and if EC and AF
intersect in B, then OA BC + OC • AB + OB • CA = 0. If 0, A, B, C
are concyclic this gives Ptolemy's theorem.

8. If ABC is a triangle and LM a segment, and if we construct
LMP similar to ABC, LMQ similar to BCA, and LMR similar to CAB,
then PQR is similar to CAB.

9. If the variable complex number u depends on the real number x
as a variable parameter, by the linear fractional form

ax + b

u -

ex + d

then for different values of x the vector representing u will terminate on
a circle.

For if we construct

b

U~d '

w =

a

u

c

70 VECTOR CALCULUS

this reduces to — (cx/d), hence the angle of w, which is the angle between
u — ale and u — b/d, is the angle of — d/c and is therefore constant.
Hence the circle goes through a/c (x = «) and b/d (x = 0).

10. If

_ x(c — b)a + b(a — c) ,
U~ k(c-b) + (a- c)

where x is a variable real parameter, then the vector representative of
u will terminate on the circle through A, B, C, where OA represents
a, OB represents 6, and OC represents c.

11. Given three circles with centers C1} d, C3, and O their radical
center, P any point in the plane, then the differences of the powers of
P with respect to the three pairs of circles are proportional respectively
to the projections of the sides of the triangle CiC2Cz on OP.

12. Construct a polygon of n sides when there is given a set of points,
Ci, C2, - • -, Cn which divide the sides in given ratios ax : bi, a2 : 62, • • •,
a» : 6„.

If the vertices are &, £2, • • • , in, and the points Ci, C2, • • • , Cn are
at the ends of vectors 71, 72, •••, yn, we have

Olll + &lfc = 7l(ai +6l) ' * * CLntn + bnh = 7n(an + bn).

The solution of these equations will locate the vertices. When is the
solution ambiguous or impossible?

13. Construct two directly similar triangles whose bases are given
vectors in the plane, fixed in position, so that the two triangles have a
common vertex.

14. Construct the common vertex of two inversely similar triangles
whose bases are given.

15. Construct a triangle ABC when the lengths of the sides AB and
AC are given and the length of the bisector AD.

1G. Construct a triangle XYZ directly similar to a given triangle
PQR whose vertices shall be at given distances from a fixed point 0.

Let the length of OX be a, of OY be 6, and of OZ be c. Then X is
anywhere on the circle of radius a and center O. We have XY/XZ
= PQIPR, that is,

OY -OX = PQ
OZ-OX PR'
whence we have

OXQR + OYRP + OZPQ = 0.
We draw OXK directly similar to RPQ giving KO/OX = QR/RP and

KO + OY + OZ -£§ = 0, that is,

VECTORS IN A PLANE 71

In KOY we have the base KO and the length OY = b, and length of

_ length PQ
length RP'

We can therefore construct KOY and the problem is solved.

17. The hydrographic problem. Find a point X from which the three
sides of a given triangle ABC are seen under given angles.

XB/XA = y cks 0, XC/XA = z cks p.

XB = XA + AB, XC = XA + AC.
Eliminate XA giving 2 cks <?•# A + y cksd-AC = BC. Find U such
that z AJBI7 = Z AXC, Z ACtf = Z AXB, then BU = z cks *>.
BA,CU = y cks OCA.

Construct A ACX directly similar to A A UB.

18. Find the condition that the three lines perpendicular to the
three vectors pa, qa, ra at their extremities be concurrent.

We have p + xkp = q + ykq = r + zkr. Taking conjugates
q — xkp = p — ykq = r— zkr. Eliminate x, y, z from the four
equations.

19. If a ray at angle 0 is reflected in a mirror at angle a the reflected
ray is in the direction whose angle is 2 a — /3. Study a chain of mirrors.
Show that the final direction is independent of some of the angles.

20. Show that if the normal to a line is a and a point P is distant y
from the line, and from P as a source of light a ray is reflected from the
line, its initial direction being — qa, then the reflected ray has for
equation — 2ya + tqa = p.

For further study along these lines, see Laisant: Theorie et
Application des Equipollences.

11. Alternating Currents. We will notice an application
of these hypernumbers to the theory of alternating currents
and electromotive forces, due to C. P. Steinmetz.

If an alternating current is given by the equation

I = Io cos 2wf(t - h),

the graph of the current in terms of t is a circle whose
diameter is 70 making an angle with the position for t = 0
of 2wfti. The angle is called the phase angle of the current.
If two such currents of the same frequency are superim-

72 VECTOR CALCULUS

posed on the same circuit, say

we may set

7 = 70 cos 2irf(t - ti),
F = Jo' cos 2tt/(* - fcO,
sex

70 cos 2vfh + h' cos 2tt/V = 70" cos 2wfh,

70 sin 2tt/<i + U sin 2u//i' = 70" sin 2irft2t

7" = 70" cos 2wf(t - it),

which also has for its graph a circle, whose diameter is the
vector sum of the diameters of the other two circles. We
may then fairly represent alternating currents of the simple
type and of the same frequency by the vectors which are
the diameters of the corresponding circles. The same
may be said of the electromotive forces.

If we represent the current and the electromotive force
on the same diagram, the current would be indicated by a
yellow vector (let us say) traveling around the origin,
with its extremity on its circle, while at the same time the
electromotive force would be represented by a blue vector
traveling with the same angular speed around a circle
with a diameter of different length perhaps. The yellow
and the blue vectors would generally not coincide, but they
would maintain an invariable angle, hence, if each is con-
sidered to be represented by a vector, the ratio of these
vectors would be such that its angle would be the same for
all times. This angle is called the angle of lag, or lead,
according as the E.M.F. is behind the current or ahead of it.

The law connecting the vectors is

E= ZI,

where E is the electromotive force vector, that is, the vector
diameter of its circle, 7 is the current vector, the diameter
of its circle, and Z is a hypernumber called the impedance,

VECTORS IN A PLANE 73

[<p/0], measured in ohms. The scalar part of Z is the
resistance of the circuit, while the imaginary part is the
reactance, the formula for Z being

Z = r — xk.

The value of x is 2irfL, where/ is the frequency, [T~l], and
L is the inductance, [^G-1?1], in henry s, or — l/2irfC where
C is the permittance, [OT1^-1], in farads. [1 farad = 9- 1011
e.s. units = 10-9 e.m. units, and 1 ^nn/ = ^lO-11 e.s.
units = 109 e.m. units.] It is to be noticed that reactance
due to the capacity of the circuit is opposite in sign to
that due to inductance.

The law above is called the generalized Ohm's law. We
may also generalize KirchofFs laws, the two generalizations
being due to Steinmetz, and having the highest importance,
inasmuch as by the use of these hypernumbers the same
type of calculation may be used on alternating circuits as
on direct circuits. The generalization of KirchofFs laws
is as follows :

(1) The vector sum of all electromotive forces acting in a
closed circuit is zero, if resistance and reactance electro-
motive forces are counted as counter electromotive forces.

(2) The vector sum of all currents directed toward a
distributing point is zero.

(3) In a number of impedances in series the joint im-
pedance is the vector sum of all the impedances, but in a
parallel connected circuit the joint admittance (reciprocal
of impedance) is the sum of the several admittances.

The impedance gives the angle of lag or lead, as the angle
of a hyper number of this type.

We desire to emphasize the fact that in impedances we
have physical cases of complex numbers. They involve
complex numbers just as much as velocities involve positive

74 VECTOR CALCULUS

of negative velocity, or rotations involve positive or nega-
tive. We may also affirm that the complex currents and
electromotive forces are real physical existences, every
current implying a power current and a wattless current
whose values lag 90° (as time) behind the power current.
The power electromotive force is merely the real part of
the complex electromotive force, and the wattless E.M.F. the
imaginary part of the complex electromotive force, both
being given by the complex current and the complex
impedance.

We find at the different points of a transmission line that
the complex current and complex electromotive force satisfy
the differential equations

dl/ds = (g + Cok)E, dE/ds = (r + Look)L

The letters stand for quantities as follows: g is mhos I mile,
r is ohms/mile, C is farads/mile, L is henrys/mile. co = 2irf.
Setting

m* = (r + Lo>k)(g + Cirk), I2 = (r + Lak)/(g + Cwk),
so that m is [X-1] while / is ohms/mile, the solution of the
equations is

E = E0 cosh ms + ll0 sinh ms,
I = Iq cosh ms + 1~1Eq sinh ms,

where E0 and 70 are the initial values, that is, where s = 0.
If we set Eq = ZqIq and then set Z0 = Z cosh h, I =
Z sinh h we have

E = Z cosh (ms + h)I0, I = l~lZ sinh (ms + h)I0,

E = I coth (ms + h)I,
E = sech h cosh (ms + h)E0,
I = csch h sinh (ms + h)I0.

To find where the wattless current of the initial station has
become the power current we set I = kl0, that is,

sinh (ms -f- h) = k sinh h.

VECTORS IN A PLANE 75

The value of s must be real.

EXAMPLES
(1) Let r = 2 ohms/mile, L = 0.02 henrys/mile,

C = 0.0000005 farads/mile,
g = 0, to = 2000, coL = 40 ohms/mile, conductor

reactance,
r + Look = 2 + 40/c ohms/mile impedance

= 40.587.i5o.
uC = 0.001 mhos/mile dielectric susceptance.
g + Coik = 0.001 k mhos/mile dielectric admit-
tance = 0.00190°.
(g + Cuk)~l = 1000/j"1 = 100027o° ohms/mile

dielectric impedance.
m2 = 0.0405i77.i5°, m = 0.200188.58°,
P = 40500_.2.85°, I = 201.25_i.43°.

Let the values at the receiver (s = 0) be

E0 = 10000o volts, 70 = 00o.
Then we have E = 1000 cosh s0.200188.58°,

for s = 100 E = 1000 cosh 20.0188.58 = 625.945.92°,

I = 2.7727o,
for s = 8 E = 50.01i26.ot,
for s = 16 E = 1001i80.3°,
for s = 15.7 E = 1000i8o°, a reversal of phase,
for s = 7.85 E = 090o.

At points distant 31.4 miles the values are the same.
If we assume that at the receiver end a current is to be
maintained with

Jo = 5040° with E0 = 10000o,

E = 1000 cosh s0.200188.58° + 1006238.57° sinh s0.200188.58°,
I = 504o° cosh sm + 5i.43° sinh sm.
At s = 100 E = 10730n355°.

MacMahon, Hyperbolic Functions.

76 VECTOR CALCULUS

(2) Let E0 - 10000, 70 - 65i3.5° r = 1, g = 0.00002

Ceo = 0.00020 period 221.5 miles, o>L = 4.

(3) The product P = EI represents the power of the
alternating current, with the understanding that the fre-
quency is doubled. The real or scalar part is the effective
power, the imaginary part the wattless or reactive power.
The value of TP is the total apparent power. The cos z P
is the power factor, and sin / P is the induction factor.
The torque, which is the product of the magnetic flux by
the armature magnetomotive force times the sine of their
angle is proportional to TIP, where E is the generated
electromotive force, and/ is the secondary current. In
fact, the torque is TI'EI-p/2irf where p is the number of
poles (pairs) of the motor.

12. Divergence and Curl. In a general vector field the
lines have relations to one another, besides having the
peculiarities of the singularities of the field. The most
important of these relations depend upon the way the lines
approach one another, and the shape and position of a
moving cross-section of a vector tube. There is also at
each point of the field an intensity of the field as well as a
direction, and this will change from point to point.

Divergence of Plane Lines. If we examine the drawing
of the field of a vector distribution in a plane, we may
easily measure the rate of approach of neighboring lines.
Starting from two points, one on each line, at the intersec-
tion of the normal at a point of the first line and the second
line, we follow the two lines measuring the distance apart
on a normal from the first. The rate of increase of this
normal distance divided by the normal distance and the
distance traveled from the initial point is the divergence of
the lines, or as we shall say briefly the geometric divergence
of the field. It is easily seen that in this case of a plane

VECTORS IN A PLANE 77

field it is merely the curvature of the curves orthogonal
to the curves of the field.

For instance, in the figure, the tangent to a curve of the
field is a, the normal at the same point /5. The neighboring
curve goes through C. The differential of the normal,
which is the difference of BD and
AC, divided by AC, or BD, is the
rate of divergence of the second curve
from the first for the distance AB.
Hence, if we also divide by AB we
will have the rate of angular turn of
the tangent a in moving to the neigh-
boring curve, the one from C. This rate of angular turn
of the tangent of the field is the same as the rate of turn of
the normal of the orthogonal system, and is thus the curva-
ture of the normal system.

Curl of Plane Lines. If we find the curvature of the
original lines of the field we have a quantity of much im-
portance, which may be called the geometric curl. This
must be taken plus when the normal to the field on the
convex side of the curve makes a positive right angle with
the tangent, and negative when it makes a negative right
angle with the tangent. Curl is really a vector, but for
the case of a plane field the direction would be perpendicular
to the plane for the curl at every point, and we may con-
sider only its intensity.

Divergence of Field. Since the field has an intensity as
well as a direction, let the vector characterizing the field
be cr = Ta-a. Then the rate of change of TV in the direc-
tion of a, the tangent, is represented by daT<r. Let us
now consider an elementary area between two neighboring
curves of the field, and two neighboring normals. If we
consider Ta as an intensity of some quantity whose amount

78 VECTOR CALCULUS

depends also upon the length of the infinitesimal normal
curve, so that we consider the quantity Ta-dn, then the
value of this quantity, which we will call the transport of
the differential tube (strip in the case of a plane field),
TV being the density of transport, will vary for different
cross-sections of the tube, and for the case under considera-
tion, would be Ta'dn' - Tadn. But TV' = TV + daTa-ds
and dn' = dn + ds-dn times the divergence of the lines.
Therefore, the differential of the transport will
P" ~T~( be (to terms of the first order) ds X dn X ( TV

I L— times divergence + daTa). Hence, the density

F ' of this rate of change of the transport is TV
. times the divergence + the rate of change of TV
along the tangent of the vector line of the field. This quan-
tity we call the divergence of the field at the initial point, and
sometimes it will be indicated by div. cr, sometimes by
— SVa, a notation which will be explained. It is clear
that if the lines of a field are perpendicular to a set of straight
lines, since the curvature of the straight lines is zero, the
divergence of the original lines is zero, and the expression
reduces to daT<r.

Curl of Field. We may also study the circulation of the
vector a along its lines, by which we mean the product of
the intensity TV by a differential arc, that is, Tads. On
the neighboring vector line there is a different intensity,
TV', and a different differential arc ds'. The differential
of the circulation is easily found in the same manner as
the divergence, and turns out to be

— (dfiTa + TV X curl of the vector lines).

This quantity we shall call the curl of the field, written
sometimes curl a, and more frequently Wa, which notation
will be explained.

1

VECTORS IN A PLANE 79

It is evident that the curl of a is the line integral of the
Tads around the elementary area, for the parts contributed
by the boundary normal to the field will be zero. Hence,
we may say that curl a is the limit of the circulation of <r
around an elementary area constructed as above, to the
area enclosed. We will see later that the shape of the area
is not material.

Likewise, the divergence is clearly the ratio to the elemen-
tary area of the line integral of the normal component of <r
along the path of integration. We will see that this also
is independent of the shape of the area.

Further, we see that in a field in which the intensity of a
is constant the divergence becomes the geometric divergence
times the intensity TV, and the curl becomes the geometric
curl times the intensity T<r.

Divergence and curl have many applications in vector
analysis in its applications to geometry and physics. These
appear particularly in the applications to space. A simple
example of convergence or divergence is shown in the
changing density of a gas moving over a plane. A simple
caSfc of curl is shown by a needle imbedded in a moving
viscous fluid. The angular rate of turn of the direction of
the needle is one-half the curl of the velocity.

13. Lines as Levels. If the general equation of a given
set of curves is

u(x, y) = c,

the§e curves will be the vector lines of an infinity of fields,
for if the differential equation of the lines is

dx/X m dy/Y,
then we must have

Xdu/dx + Ydu/dy = 0
and for the field

a = Xa + Y0.

80 VECTOR CALCULUS

We may evidently choose X arbitrarily and then find Y
uniquely from the equation. However, if a\ is any one
field so determined, any other field is of the form

a = <TiR(x, y).

The- orthogonal set of curves would have for their finite
equation

v(x, y) = c

and for their differential equation

Xdvldy - Ydv/dx = 0.

If we use a uniformly to represent the unit tangent of
the u set, and P the unit tangent of the v set, then P = ha.
The gradient of the function u is then d0u-(3, and the
gradient of the function v is — dav-a. But the gradient
of u is also (ux, uy) and of v is (vx, vv) = (uVf — ux). It
follows that the tensors of the gradients are equal. In fact,
writing Vm for gradient u, we have Vt> = kVu. We also
have for whatever fields belong to the two sets of orthog-
onal lines for u curves, a = rVv, for the v curves, a' = sVu,
or also we may write

Vv = tot, Vu = tp, a = Ta-ct.

14. Nabla. The symbol V is called nabla, and evidently
may be written in the form ad/dx + Pd/dy for vectors in
a plane. We will see later that for vectors in space it
may be written ad/dx + Pd/dy + yd/dz, where a, ft y are
the usual unit vectors of three mutually perpendicular
directions. However, this form of this very important
differential operator is not at all a necessary form. In
fact, if a and fi are any two perpendicular unit vectors in
a plane, and dr, ds are the corresponding differential dis-
tances in these two directions, then we have

V = ad/dr + pd/ds.

VECTORS IN A PLANE 81

For instance, if functions are given in terms of r, 6, the
usual polar coordinates, then V = Upd/dr + kUpd/rdd.
The proof that for any orthogonal set of curves a similar
form is possible, is left to the student. In general, V is
defined as follows : V is a linear differentiating vector
operator connected with the variable vector p as follows:
Consider first, a scalar function of p, say F(p). Differentiate
this by giving p any arbitrary differential dp. The result
is linear in dp, and may be looked upon as the product of
the length of dp and the projection upon the direction of
dp of a certain vector for each direction dp. If now these
vectors so projected can be reduced to a single vector,
this is by definition VF. For instance, if F is the distance
from the origin, then the differential of F in any direction
is the projection of dr in a radial direction upon the direc-
tion of differentiation. Hence, V7p = Up. In the case
of plane vectors, VF will lie in the plane. In case the
differential of F is polydromic, we define VF as a poly-
dromic vector, which amounts to saying that a given set
of vectors will each furnish its own differential value of dF.
In some particular regions, or at certain points, the value
of J7F may become indefinite in direction because the
differentials in all directions vanish. Of course, functions
can be defined which would require careful investigation
as to their differentiability, but we shall not be concerned
with such in this work, and for their adequate treatment
reference is made to the standard works on analysis.

We must consider next the meaning of V as applied to
vectors. It is evident that if V is to be a linear and there-
fore distributive operator, then such an expression as Va
must have the same meaning as VXa. + V Y(3 + VZy if
a = Xa + F/3 -r Zy, where a, 0, y are any independent
constant vectors. This serves then as the definition of

82 VECTOR CALCULUS

Vo-, the only remaining necessary part of the definition is
the vector part which defines the product of two vectors.
This will be considered as we proceed.

15. Nabla as a Complex Number. We will consider now
p to represent the complex number x + yk, or re, and that
all our expressions are complex numbers. The proper
expression for V becomes then

V = d/dx + kd/dy = Upd/dr + kUpd/rdd.

In general for the plane, let p depend upon two parameters
u, v, and let

dp = p\du -f- p2dv.

If a is a function of p (generally not analytic in the usual
sense) and thus dependent on u, v, we will have

da = dcr/du-du + da/dv-dv = R-dpV -a.

If we multiply dp by kpi, which is perpendicular to pi, the
real part of both sides will be equal and we have, since kpi
is perpendicular to pi,

Rkpidp — dvRkpip2,
and similarly

Rkpidp = duRkpipi = — duRkpip2

since the imaginary part of pip2 equals — the imaginary part
of p2pi-

Substituting in da we have

A, = «.*,(-,*£- £+#-£) <r.

\ Rkpip2ou Rkpip2dvJ

The expression in (), however, is exactly what we have de-
fined above as V, and thus we have proved that we may
write V in the form corresponding to dp in terms of u and v :

V = k(p2d/du — pid/dv)/Rkpip2.

In case pi and p2 are perpendicular the divisor evidently

VECTORS IN A PLANE 83

reduces to ± Tp\Tp2 according as p2 is negatively perpendic-
ular to pi or positively perpendicular to it. We may write
V in this case in the form (since p2 = — kpi- Tp2/Tpi or
+ kpr Tp2/Tpi)

v = _pi_A . _Pi_ A = pfii,p-i 1_ .

TPl2du^ TP22dv F du^Ht dv

In any case we have dF = Rdp\/F, da = Rdp\7 -v.

Also in any case V = Vu-d/du + \7v-d/dv.

16. Curl, Divergence, and Nabla. Suppose now that a
is the complex number for the unit tangent of one of a set
of vector lines, and |8 the complex number for the unit
tangent of the orthogonal set, at the same point. The
curvature of the orthogonal set is the intensity of the vector
rate of change of (3 along the orthogonal curve. But this
is the same as the rate of change of the unit tangent a as
we pass along the orthogonal curve from one vector line to
an adjacent one. The differential of a is perpendicular
to a, and hence parallel to the direction of /3. Hence this
curvature can be written

But if we also consider the value of R- a(R-aV)a, since the
differential of a in the direction of a has no component
parallel to a, this term is zero, and may be added to the
preceding without affecting its value. Hence the curvature
of the orthogonal set reduces to

R(aRaV + ^/3V)« = R-Va.

This is the divergence of the curves of a.

If now <j = Tcr-a, we find from the definition of the
divergence of a that it is merely

R-Va.
Considering in the same manner the definition of curl of a,

84 VECTOR CALCULUS

we find it reduces to — R-kV<r, and if we multiply this by k,
so that we have

curl a = - kRW(T=LV<rf

we see at once that when added to the expression for the
divergence of a we have

div-<7 + curl <r = V<r.

The real part of this expression is therefore the divergence
of a, and the imaginary part is the curl of a. This will
agree with expressions for curl and divergence for space of
three dimensions. We have thus found some of the
remarkable properties of the operator V .

17. Solenoidal and Lamellar Vector Fields. When the
divergence of a is everywhere zero, the field is said to be
solenoidal. If the curl is everywhere zero, the field is called
lamellar.

18. Properties of the Field. Let a set of curves u = c be
considered, and the orthogonal set v — a, and let the field a
be expressed in the form

o- = XVu + FVfl,

where it is assumed that the gradients Vu, Vv exist at all
points to be considered. We have then

diver = RVa = RvXVu+ RvYVv _

+ XRWu+ YRWv.

The expression RWu is called the plane dissipation of u.
In case it vanishes it is evident that u satisfies Laplace's
equation, and is therefore harmonic.
We also have

curl o- = I- V<r = — kRkVXVu — kRkvYVv,

the other parts vanishing.

VECTORS IN A PLANE 85

Since we have chosen orthogonal sets of curves we may
write these in the forms

diver = (TVu)2dX/du + (TVv)2dY/dv

+ XRVVu + YRvVv,
curl o- = (TVu)(TW)(dY/du - dX/dv)k.

In case we have chosen the lines of cr for the u curves,
then X = 0, and a = Y V v

diver = YRVW+ (TVv)2dY/dv,
curl (7= TVuTVvdY/du-k.

We notice that curl Vu = 0, curl Vv = 0, div k\/u — 0,
divkVv = 0, kVu = VvTVu/TVv, and for

Y = TVu/TVv,
we have

(TVu)-2RvS7u = d log (TVu/TVv)/du,
' (fV*)~*BVV« = d log (TVv/TVu)/dv.

We may now draw some conclusions as to the types of
curves and <r. (Cf. B. O. Peirce, Proc. Amer. Acad. Arts
and Sci., 38 (1903) 663-678; 39 (1903) 295-304.)

(1) The field will be solenoidal if diver = 0, hence

d log Y/dv = - RVW/TW2, '
which may be integrated, giving

Y = ef(u' v) + o{u).

If v is harmonic, Y is a function of u only and a =G(u)Vv.

(2) If the field is lamellar, curl a = 0, and Y is a function
of v only, so that a = H(v)Vv = VL(v).

(3) If the field is both solenoidal and lamellar,

RVVL(v) = 0, whence RVVv/(TVv)2 = /(*),
which is a condition on the character of the curves. Hence

86 VECTOR CALCULUS

it is not possible to have a solenoidal and lamellar field
with purely arbitrary curves.

(4) If the field is solenoidal and Ta, the intensity,
is a function of u alone, Y = p(u)/TVv, and therefore
d log Y/dv = - dTVv/TVvdv = - RvVv/TVv2, whence

2RVW = d(TVv)2/dv,

which is a condition on the curves. An example is the
cross-section of a field of magnetic intensity inside an in-
finitely long cylinder of revolution which carries lengthwise
a steady current of electricity of uniform current density.

(5) If a is lamellar and Ta is a function of v only, TVv
= g(v). An example is the field of attraction within a
homogeneous, infinitely long cylinder of revolution. The
condition is a restriction on the possible curves.

(6) If the field is lamellar and Ta a function of u only,
since Y is a function of v only, d log TVv/du = k(u), or
TVv = l(u)/m(v).

This restricts the curves.

(7) If the field is solenoidal and Ta a function of v only,
Ta = p(v)TW. Therefore d log Ta/dv = d log TVa/dv
— RS7Vv/(T\7v)2. Hence either both sides are constant
or else both expressible in terms of v. If the field is not
lamellar also, TVv must then be a function of u as well as
of v.

(8) If the field is lamellar and has a scalar potential
function, that is, a = VP, then since a = q(v)Vv, we must
have P a function of v only, and a = P'Vfl. From this
it follows that diver = P\v)RVVv + P"(v)(TVv)2.

(9) If the field is uniform, Ta — a, Y = a/T\7v, and a
is lamellar only if TVv is either constant or a function of
v only, while a is solenoidal only if we have

2RVW = d(TVv)2/dv.

VECTORS IN A PLANE 87

(10) Whatever function u is, the u lines are vector lines
for the vectors £ = f(u)UVv, f = g(v)U\7v, or

T? = *(«, r)tTVf.

(11) If the field is solenoidal, TV a function of u only,
and the w curves are the lines of the field, then the curl
takes the form — k div • ka, whence it has the form

k[b(u)RVVu+ b'(u)(TVu)2],

where b may be any differentiate function. If TV is also
a function of v, the form of the curl is

k[b(u, v)RVVu + db(u, v)/du(TVu)2].

(12) If TV is a function of u only, the divergence takes
the form

diver - Ta[RWv/TVv - dTVv/dv].

(13) If TV is a function of v only

curl a = - kTaTVu/TVv-dTVv/du.

19. Continuous Media. When the field is that of the
velocity of a continuous medium, we have two cases to
take into account. If the medium is incompressible it is
called a liquid, otherwise a gas. Incompressibility means
that the density at a point remains invariable, and if this
is c, then from

dc/dt= dc/dt + RaVc, = dc/dt + RV(ca) - cRV<r
we see that the first two terms together vanish, giving the
equation of continuity, since they give the rate per square
centimeter at which actual material (density times area,
since the height is constant) is changing. Hence in this case

dc/dt = — cRV<t>

This gives the rate of change of the density at a point
moving with the fluid. Hence if it is incompressible, the
velocity is solenoidal, RV& = 0.

88 VECTOR CALCULUS

This may also be written curl (— ka) = 0, hence — ka
= V?, and <j — kvQ, which shows that for every liquid
there is a function Q called the function of flow.

When curl { = 0, we have seen that £ is called lamellar.
It may also be called irrotational, since the curl is twice the
angular rate of rotation of the infinitesimal parts of the
medium, about axes perpendicular to the plane, and if
curl { = 0 there is no such rotation. Curl is analogous
to density, being a density of rotation when the vector
field is a velocity field.

The circulation of the field is the integral fRadp along
any path from a point A to a point B. This is the same as
Xdx + Ydy, and is exact when

dX/dy = dY/dx.

But this gives exactly the condition that the curl should
vanish. Hence if the motion is irrotational the circulation
from one point to another is independent of the path. In
this case we may write a = VP where P is called the
velocity potential.

When a is irrotational, the lines of Q have as orthogonals
the lines of P. If the motion is rotational, these orthogonals
are not the lines of such a function as P. If the motion is
irrotational, we have for a liquid, RwP = 0, and P must
be harmonic. Hence if the orthogonal curves of the Q
curves can belong to a harmonic function they can be curves
of a velocity potential. If a set of curves belong to the
harmonic function u, then RWu = 0, and this shows that
the curl of — JcVu is zero, whence Rdp(— k\/u) is exact
= dv, where Vv = — kVu. From this we have Vm
= kvQ for the condition that the orthogonal curves belong
to a harmonic function. This however gives the equation
TS/u = TvQ. We may assert then for a liquid that there
is always a function of flow, and the curves belonging to

VECTORS IN A PLANE 89

this function are the vector lines of the velocity, the in-
tensity of the velocity being the intensity of the gradient of
the function of flow. If the orthogonal curves belong to
a function which has a gradient of the same intensity, both
functions are harmonic, the function of the orthogonal set
is a velocity potential, and the motion is irrotational.

We have a simple means of discovering the sets of curves
that belong to harmonic functions, as is well known to
students of the theory of functions of a complex variable,
since the real and the imaginary part of an analytic function
of a complex variable are harmonic for the variable co-
ordinates of the variable. That is to say, if p = x + yk,
and £ = /(p) = u -\- vk, then u, v are harmonic for x, y.
The condition given by Cauchy amounts to the equation
Vm = — k\/v, or V£ = 0 where £ is a complex number.
It is clear from this that the field of £ is both solenoidal and
lamellar, a necessary and sufficient condition that £ be an
analytic function of a complex variable. In this case £ is
called a monogenic function of position in the plane. It is
clear that £ = VH where H is a harmonic function.

In case there are singularities in the field it is necessary
to determine their effect on the integrals. For instance, if
we have a field a and select a path in it, from A to B, or a
loop, the flux of a through the path is the integral of the
projection of a on the normal of the path, that is, if the path
is a curve given by dp, so that the projection is Ra(— kdp),
the integral of this is the flux through, the path. It is
written

2 = SI (- Rakdp) = - kfladp.

In the case of a liquid the condition RV<r = 0 shows that
the expression is integrable over any path from A to B,
with the same value, unless the two paths enclose a singu-
larity of the field. In the case of a node, the integral around

7

90 VECTOR CALCULUS

a loop enclosing the node is called the strength of the source
or sink at the node. We may imagine a constant supply
of the liquid to enter the plane or to leave it at the node,
and be moving along the lines of the field. Such a system
was called by Clifford a squirt

If the circulation is taken around a singular point it
will usually have a different value for every turn around the
point, giving a polydromic function. These peculiarities
must be studied carefully in each case.

EXERCISES

1. From £ = Apn we find in polar coordinates that

u = Arn cos nd, v = Arn sin nd.

These functions are harmonic and their curves orthogonal. Hence
if we set a = Vwora = V#, we shall have as the vector lines of <r the v
curves or the u curves. What are the curves for the cases n = — 3,
— 2, — 1, 1, 2, 3? What are the singularities?

2. Study £ = A log p, and £ = A log (p — a)/(p + a).

3. Consider the function given implicitly by p = £ + e*. This
represents the flow of a liquid into or out of a narrow channel, in the
sense that it gives the lines of flow when it is not rotational.

4. Show that a = A/p gives a radial irrotational flow, while a = Ak/p
gives a circular irrotational flow. What is true of a = Akpl The
last is Clifford's Whirl.

5. Study a flow from a source at a given point of constant strength
to a sink at another point, of the same strength as the source.

6. If the lines are concentric circles, and the angular velocity of any
particle about the center is proportional to the n-th power of the radius
of the path of the point, show that the curl is \ {n + 2) times the angular
velocity.

7. A point in a gas is surrounded by a small loop. Show that the
average tangential velocity on the loop has a ratio to the average
normal velocity which is the ratio of the tensor of the curl to the
divergence.

8. What is the velocity when there is a source at a fixed origin, and
the divergence varies inversely as the w-th power of the distance from
the origin. [The velocity potential is A log r — B{n — 2)~2r2-n.]

9. Consider the field of two sources of equal strength. The lines are
for irrotational motion, cassinian ovals, where, if r, r' are the distances

VECTORS IN A PLANE 91

from the two sources (foci) and rr' — h2, Q = A log h + B, the velocity
is such that T<r = ATp/h2, the origin being half way between the foci;
the orthogonal curves are given by u = iA[ir/2 — (0 + di)] where 0,
0i are the angles between the axis and the radii from the foci, that is
they are equilateral hyperbolas through the foci. The circulation
about one focus is ttA, about both 2irA.

10. If the lines are confocal ellipses given by

z2/m + i/VG* - c2) = 1,

then Q = A log ( \V + V (m — c2)) + B. If p is the perpendicular
from the center upon the tangent of the ellipse at any point, then the
velocity at the point is such that T<r = — Ap/ y/ [/*(/* — c2)], and
the direction of <r is the unit normal. The potential function is
A sin-1 B' V v\c. V v is the semi-major axis. What happens at the foci?

11. If the stream lines are the hyperbolas of the preceding, then
a = 2 A V (*7(m — v)) times the unit normal of the hyperbola. On the
line p = yka there is no velocity, at the foci the velocity is oo , half way
between it is 0. The lines along the major axis outside the foci act
like walls.

12. If we write for brevity ux for T\7u, and vi for T\7v, show that
we have whether the u curves are orthogonal to the v curves or not,

V V = Ui2d2jdu2 + Vi2d2ldv2 + VVud/du + VVvd/dv

+ 2RVuVvd2/dudv.

If the sets of curves are orthogonal the last term vanishes; if u and v
are harmonic the third and fourth terms drop out; if both cases happen,
only the first two terms are left.

13. In case of polar coordinates, Vr = Up, V0 = r"2A;p and

VV = d2/dr2 + r~ldldr + r_2d2/d02.

14. A gas moves in a plane in lines radiating from the origin, which
is a source. The divergence is a function of r only, the distance from
the center. Find the velocity and the density at any point.

a = pf{r), flVo- = eir) = 2/(r) + rf'{r),
and

f(r) = Ajr2 + r~2fre{r)dr.
To determine c,

RV log co- = - e(r) = f(r)Rp\/ log c = rf(r)d log c/dr.

15. Show that in the steady flow of a gas we may find an integrating
factor for Rdpka by using the density, [dc/dt = 0 = Rsjca = curl -fcco-,
and Rdpkca is exact.]

16. A fluid is in steady motion, the lines being concentric circles.
The curl is known at each point and the tensor of a is a function of r
only. Find the velocity and the divergence.

92 VECTOR CALCULUS

17. Rotational motion, that is a field which is not lamellar, is also
called vortical motion. The points at which the curl does not vanish
may be distributed in a continuous or a discontinuous manner. In
fact there may be only a finite number of them, called vortices. We
have the following:

<r = k\7Q, VVQ = T curl a = 2«,
Q = 7r_1//«' log rdx'dy' + Q0,

where «' denotes co at the variable point of the integration, r is the
variable distance from the point at which the velocity is wanted, and Q0
is any solution of Laplace's equation which satisfies the boundary
conditions.

If the mass is unlimited and is stationary at infinity we have

« = kfwfftt'ifi - P')/T(p - py-dx'dy'.

A single vortex filament at p of strength I would give the velocity

a =U2T.(p-p')IT(p-p')\

If we multiply the velocity at each point p at which there is a vortex by
the strength, and integrate over the whole field, we find the sum is zero.
There is then a center of vortices where the velocity is zero, something
like a center of gravity. Instances are

(1) A single vortex of strength I. The vortex point will remain at
rest, and points distant from it r will move on concentric circles with
the vortex as center, and velocity l/2wr. The circulation of any loop
surrounding the vortex is of course the strength.

(2) Two vortices of strengths k, U. They will rotate about the
common center of gravity of two weighted points at the fixed distance
apart a, the weights being the two strengths. The angular velocity of
each is

27ra2

The stream lines of the field are given by fxhf%h = const. When
k = — {, the center is at infinity, and the vortices remain a fixed dis-
tance apart, moving parallel to the perpendicular bisector of this segment
joining them. Such a combination is called a vortex pair. The stream
lines of the accompanying velocity are coaxal circles referred to the
moving points as limit points. The plane of symmetry may be taken
as a boundary since it is one of the stream lines, giving the motion of a
single vortex in a field bounded by a plane, the linear velocity of the
vortex being parallel to the wall and \ of the velocity of the liquid along
the wall. The figure suggests the method of images which can indeed
be applied. For further problems of the same character works on
Hydrodynamics should be consulted.

VECTORS IN A PLANE 93

18. Liquid flows over an infinite plane towards a circular spot where
it leaks out at the rate of 2 cc. per second for each cm.2 area of the leaky
portion. The liquid has a uniform depth of 10 cm. over the entire
plane field. Find formulas for the velocity of the liquid inside the
region of the leaky spot, and the region outside, and show that there is
a potential in both regions.

a = iVp in spot, 40/p outside, P = ^pp in spot, 40 log Tp — 20 log
400 outside.

Find the flux through a plane area 20 cm. long and 10 cm. high, whose
middle line is 5 cm. from the center of the leaky spot, also when it is
30 cm. from the leaky spot. Find the divergence in the two regions.

Franklin, Electric Waves, pp. 307-8.

19. Show that in an irrotational motion with sources and sinks, the
lines of flow are the orthogonal curves of the stream lines of a correspond-
ing field in which the sources and the sinks are replaced by vortices of
strengths the same as that of the sources and sinks, and inversely.
Stream lines and levels change place as to their roles. For sources and
sinks Q = 1/2tt-ZZi0i, P = 1/2* -Z log rxh.

20. Vector Potential. In the expression a = — VkQwe
express (rasa vector derived by the operation of V upon
— JcQ, the latter being a complex number. In such a case
we may extend our terminology and call — JcQ the vector
potential of a. A vector may be derived from more than
one vector potential. In order that there be a vector
potential it is necessary and sufficient that the divergence
of <t vanish. Hence any liquid flow can have a vector
potential, which is indeed the current function multiplied
by — k. It is clear that Q must be harmonic.

CHAPTER VI

VECTORS IN SPACE

1. Biradials. We have seen that in a plane the figure
made up of two directed segments from a vertex enables
us to define the ratio of the two vectors which constitute
the sides when the figure is in some definite position. This
ratio is common to all the figures produced by rotating the
figure about a normal of the plane through its vertex, and
translating it anywhere in the plane. We may also reduce
the sides proportionately and still have the same ratio.
The ratio is a complex number or, as we will say in general,
a hypernumber.

If now we consider vectors in space of three dimensions,
we may define in precisely the same manner a set of hyper-
numbers which are the ratios of the figures we can produce
in an analogous manner. Such figures will be called
biradials. To each biradial there will correspond a hyper-
number. Besides the translation and the rotation in the
plane of the two sides of the biradial, we shall also permit
the figure to be transferred to any parallel plane. This
amounts to saying that we may choose a fixed origin, and
whatever vectors we consider in space, we may draw from
the origin two vectors parallel and equal to the two con-
sidered, thus forming a biradial with the origin as vertex.
Then any such biradial will determine a single hyper-
number. Further the hypernumbers which belong to the
biradials which can be produced from the given biradial
by rotating it in its plane about the vertex will be con-
sidered as equal.

94

VECTORS IN SPACE 95

The hypern umbers thus defined are extensions of those
we have been using in the preceding chapter, the new
feature being the different hypernumbers k which we now
need, one new k in fact for each different plane through the
given vertex. This gives us then a double infinity of
hypernumbers of the complex type, r-cks 6, where the
double infinity of k's constitute the new elements.

2. Quaternions. The hypernumbers we have thus de-
fined metrico-geometrically involve four essential param-
eters in whatever way they are expressed, since the
biradials involve two and the plane in which they lie two
more. Hence they were named by Hamilton Quaternions.
In order to arrive at a fuller understanding of their prop-
erties and relations, we will study the geometric properties
of biradials.

In the first place if we consider any given biradial, there
is involved in its quaternion, just as for the complex number
in the preceding chapter, two parts, a real part and an
imaginary part, and we can write the quaternion in the
form

q = r cos 6 + r sin 6 -a,

where a corresponds to what was written k in the preceding
chapter, and is a hypernumber determined solely by the
plane of the biradial. On account of this we may properly
represent a by a unit normal to the plane of the biradial,
so taken that if the angle of the biradial is considered to be
positive, the direction of the normal is such that a right-
handed screw motion turning the initial vector of the
biradial into the terminal vector in direction would in-
volve an advance along the normal in the direction in
which it points. It is to be understood very clearly that
the unit vector a and the hypernumber a are distinct
entities, one merely representing the other. The real

96 VECTOR CALCULUS

part of q is called, according to Hamilton's terminology, the
scalar part of q, and written Sq. The imaginary part is
called, on account of the representation of a as a vector,
the vector part of q and written Vq. The unit a is called the
unit vector of q and written UVq. The angle of q is 0 and
written Zq. The number r which is the ratio of the
lengths of the sides of the biradial is called the tensor of q,
and written Tq. The expression cos 6 + sin 6 -a = cas-d
is called the versor of q, and written Z7^.

Sq is a quaternion for which 0 = 0° or 180°, Fg is a
quaternion for which 0 = 90° or 270°. Tq is a quaternion
of 0°, being always positive, a is a quaternion of 0 = 90°,
and sometimes called a right versor.

3. Sum of Quaternions. In order to define the sum of
two quaternions we define the sum of two biradials first.
This is accomplished by rotating the two biradials in their
planes until their initial lines coincide, and then diminishing
or magnifying the sides of one until the initial vectors are
exactly equal and coincide. This is always possible. We
then define as the sum of the two biradials, the biradial
whose initial vector is the common vector of the two, and
terminal vector is the vector sum of the two terminal
vectors. The sum of the corresponding quaternions is
then the quaternion of the biradial sum. Since vector
addition is commutative, the addition of quaternions is
commutative.

Passing now to the scalar and vector parts of the quater-
nions, we will prove that they can be added separately, the
scalar parts like any numbers and the vector parts like
vectors.

In the figure let the biradial of q be OB/OA, of r be
OC/OA, and of q + r be OD/OA. Let the vector part of q,
Tq- sin Zq-UVq be laid off as a vector Vq perpendicular

VECTORS IN SPACE 97

to the plane of the biradial of q, and similarly for Vr.
Then we are to show that V(q + r) = Vq + Vr in the
representation and that this represents the vector part of
q + r according to the definition. It is evident that

OB = OB' + B'B, the first vector along OA, the second
perpendicular to OA. Also OC = OC" + C"& + C'C,
the first part along OA, the second parallel to B'B, and the
third perpendicular to the plane of OAB. The sum
OB + OC = OD, where OD = OB" + D"D' + D'Z), and
0Z>" - 05' + 00", D"£' = B'B + C,,(7,, D'D = C'C.

Hence the biradial of the sum is OD/OA, where the
scalar part is the ratio of OD" to OA. This is clearly the
sum of the scalar parts of q and r, and

S(q + f) = Sq+ Sr.

The vector part of the quaternion for OD/OA is the ratio
of D"D to OA in magnitude, and the unit part is repre-
sented by a unit normal perpendicular to OD" and D"D.
But D"D = B'B + C'C, and the ratio of D"D to OA equals
the sum of the ratios of B'B and C'C to OA. If then we
draw, in a plane through 0 which is perpendicular to OA,
the vector Vq along the representative unit normal of the
plane OAB, and of a length to represent the numerical
ratio of B'B to OA, and likewise Vr to represent the ratio
of C'C to OA laid off along the representative unit normal

98 VECTOR CALCULUS

to the plane OAC, because D"D is parallel to this plane,
as well as B'B and C"C, the representative unit vector of
q+ r will lie in the plane, and will be in length the vector
sum of Vq and Vr, that is V(q + r) as shown.

It follows at once since the addition of scalars is associa-
tive, and the addition of vectors is associative, and the two
parts of a quaternion have no necessary precedence, that
the addition of quaternions is associative.

4. Product of Quaternions. To define the product of
quaternions we likewise utilize the biradials. In this
case however we bring the initial vector of the multiplier
to coincide with the terminal line of the multiplicand, and
define the product biradial as the biradial whose initial
vector is the initial vector of the multiplicand, and the
terminal vector is the terminal vector of the multiplier.
In the figure, the product of the biradials OB/OA, and

Fig. 13.

OC/OB, is, writing the multiplier first,

OC/OB- OB/OA = OC/OA.

It is clear that the tensor of the product is the product of
the tensors, so that

T-qr= TqTr.
It follows that

U-qr = UqUr.

It is evident from the figure that the angle of the product
will be the face angle of the trihedral, AOC, or on a unit
sphere would be represented by the side of the spherical

VECTORS IN SPACE 99

triangle corresponding. It is clear too that the reversal of
the order of the multiplication will change the plane of
the product biradial, usually, and therefore will give a
quaternion with a different unit vector, though all the other
numbers dependent upon the product will remain the same.
However we can prove that multiplication of quaternions
is associative. In this proof we may leave out the tensors
and handle only the versors. The proof is due to Hamilton.
To represent the biradials, since the vectors are all taken
as unit vectors, we draw only an arc on the unit sphere,
from one point to the other, of the two ends of the two unit
vectors of the biradial. Thus we represent the biradial
of q by CA, or, since the biradial may be rotated in its
plane about the vertex, equally by ED. The others in-
volved are shown. The product qr is represented by FD,
from the definition, or equally by LM. What we have
to prove is that the product p • qr is the same as the product
pq-r, that is, we must prove that the arcs KG and LN are
on the same great circle and of equal length and direction.

Fig. 14.

Since FE = KH, ED - CA, HG = CB, LM = FD, the
points L, C, G, D are on a spherical conic, whose cyclic
planes are those of AB, FE, and hence KG passes through
L, and with LM intercepts on AB an arc equal to AB.
That is, it passes through N, or KG and LN are arcs of the

100 VECTOR CALCULUS

same great circle, and they are equal, for G and L are points
in the spherical conic.

5. Trirectangular Biradials. A particular pair of bira-
dials which lead to an interesting product is a pair of which
the vectors of each biradial are perpendicular unit vectors,
and the initial vector of one is the terminal of the other,
for in such case, the product is a biradial of the same kind.
In fact the three lines of the three biradials form a tri-
rectangular trihedral. If the quaternions of the three

o

Fig. 15.

are i, j, k, then we see easily that the quaternion of the
biradial OC/OB is represented completely by the unit vector
marked i, the quaternion of OA/OC by j, and of OB/OA by
k. The products are very interesting, for we have

ij = k, jk = i, hi = j,

and if we place the equal biradials in the figure we also have

ji = — k, kj = — i, ik = — j.

Furthermore, we also can see easily that, utilizing the
common notation of powers,

V- = - 1, ? - - 1, V - ■■- 1.

Since it is evidently possible to resolve the vector part of
any quaternion, when it is laid off on the unit vector of its
plane as a length, into three components along the direc-
tions of i, j, k, and since the sum of the vector parts of

VECTORS IN SPACE

10]

quaternions has been shown to be the vector part of the
sum, it follows that any quaternion can be resolved into
the parts

q = w -\- xi -\- yj -\- zk.

These hypernumbers can easily be made the base of the
whole system of quaternions, and it is one of the many
methods of deriving them. Hamilton started from these.
The account of his invention is contained in a letter to a
friend, which should be consulted. (Philosophical Maga-
zine, 1844, vol. 104, ser. 3, vol. 25, p. 489.)

6. Product of Vectors. It becomes evident at once if we
consider the product of two vector parts of quaternions,
or two quaternions whose scalar parts are zero, that we
may consider this product, a quaternion, as the product of
the vector lines which represent the vector parts of the
quaternion factors. From this point of view we ignore
the biradials completely, and look upon every geometric
vector as the representative of the vector part of a set of
quaternions with different scalars, among which one has
zero scalar. From the biradial definition we have

VqVr= S-VqVr+ V-VqVr

equal to the quaternion whose biradial consists of two
vectors in the same plane as the vector normals of the

Fig. 16.

102 VECTOR CALCULUS

biradials of Vq, Vr and perpendicular to them respectively.
In the figure the biradial of Vr is OAB, and of Vq is OBC,
and of VqVr is OAC. If then we represent the vectors by
Greek letters whether meant to be considered as lines or
as vector quaternions, a = Vq, /3 = Vr, then the quaternion
which is the product of a(3 has for its angle the angle be-
tween /3 and a + 180°, and for its normal the direction OB.
If we take UVa(3 in the opposite direction to OB, and of
unit length, so as to be a positive normal for the biradial
a /3 in that order, then we shall have, letting 6 be the angle
from a to /3,

a(3 = TaTj3(- cos 0 + UVafi sin 0).

We can write at once then the fundamental formulae

S-a& = - TaTfi cos 6, V-a$ = TaTp-sm 6- UVaP.

From this form it is clear also that any quaternion can
be expressed as the product of two vectors, the angle of
the two being the supplement of that of the quaternion,
the product of their lengths being the tensor of the quater-
nion, and their plane having the unit vector of the quater-
nion as positive normal.

If now we consider the two vectors a and 0 to be resolved
in the forms

a = ai-\- bj + ck, (3 — li + mj + nk,

where i, j, k have the significance of three mutually tri-
rectangular unit vectors, as above, then since Ta Tfi cos 6
= al-\- bm-\- en, and since the vector Ta T(3 sin 6 • UVa(3
is

(bn — cm)i + (cl — an)j + (am — bl)k,
we have

a/3 = — (al + bm + en) + (bn — cm)i

+ (cl — ari)j -\- (am — bl)k.

VECTORS IN SPACE 103

But if we multiply out the two expressions for a and 0
distributively, the nine terms reduce to precisely these.
Hence we have shown that the multiplication of vectors,
and therefore of quaternions in general, is distributive when
they are expressed in terms of these trirectangular systems.
It is easy to see however that this leads at once to the
general distributivity of all multiplications of sums.

7. Laws of Quaternions. We see then that the addition
and multiplication of quaternions is associative, that
addition is commutative, and that multiplication is dis-
tributive over addition. Multiplication is usually not
commutative. We have yet to define division, but if
we now consider a biradial as not being geometric but as
being a quaternion quotient of two vectors, we find that
P/a differs from a(3 only in having its scalar of opposite
sign, and its tensor is T(3/Ta instead of TaTfi.

It is to be noticed that while we arrived at the hyper-
numbers called quaternions by the use of biradials, they
could have been found some other way, and in fact were so
first found by Hamilton, whose original papers should be
consulted. Further the use of vectors as certain kinds of
quaternions is exactly analogous, or may be considered to
be an extension of, the method of using complex numbers
instead of vectors in a plane. In the plane the vectors
are the product of some unit vector chosen for all the plane,
by the complex number. In space a vector is the product
of a unit vector (which would have to be drawn in the
fourth dimension to be a complete extension of the plane)
by the hypernumber we call a vector. However, the use of
the unit in the plane was seldom required, and likewise in
space we need never refer to the unit 1, from which t^e
vectors of space are derived. On the other hand, just as
in the plane all complex numbers can be found as the ratios

104 VECTOR CALCULUS

of vectors in the plane in an infinity of ways, so all quater-
nions can be found as the ratios of vectors in space. All
vectors are thus as quaternions the ratios of perpendicular
vectors in space. And multiplication is always of vectors as
quaternions and not as geometric entities. In the common
vector systems other than Quaternions, the scalar part of
the quaternion product, usually with the opposite sign,
and the vector part of the quaternion product, are looked
upon as products formed directly from geometric con-
siderations. In such case the vector product is usually
defined to be a vector in the geometric sense, perpendicular
to the two given vectors. Therefore it is a function of
the two vectors and is not a number or hypernumber at
all. In these systems, the scalar is a common number, and
of course the sum of a number and a geometric vector
is an impossibility. It seems clear that the only defensible
logical ground for these different investigations is that of
the hypernumber.

It is to be noticed too that Quaternions is peculiarly
applicable to space of three dimensions, because of the
duality existing between planes and their normals. In a
space of four dimensions, for instance, a plane, that is a
linear extension dependent upon two parameters, has a
similar figure of two dimensions as normal. Hence, corre-
sponding to a biradial we should not have a vector. To
reach the extension of quaternions it would be necessary
to define triradials, and the hypernumbers corresponding
to them. Quaternions however can be applied to four
dimensional space in a different manner, and leads to a
very simple geometric algebra for four-dimensional space.
The products of quaternions however are in that case not
sufficient to express all the necessary geometrical entities,
and recourse must be had to other functions of quaternions.

VECTORS IN SPACE 105

In three-dimensional space, however, all the necessary ex-
pressions that arise in geometry or physics are easily
found. And quaternions has the great advantage over
other systems that it is associative, and that division is
one of its processes. In fact it is the most complex system
of numbers in which we always have from PQ = 0 the
conclusion P = 0, or Q = 0.*

8. Formulae. It is clear that if we reverse the order of
the product ce/3 we have

0a = Soft - Vafi.

This is called the conjugate of the quaternion a(3, and
written K-a(3. We see that

SKq = Sq= KSq, VKq = - Vq = KVq.

Further, since

qr = SqSr + SqVr + SrVq + VqVr,
we have

K-qr= SqSr - SqVr - SrVq + VrVq = KrKq.

From this important formula many others flow. We have
at once

K-qi- • -qn = Kqn> • >Kqi.
And for vectors

Koli- • -0Ln = {—)nan- • •«!.
Since

Sq = i(q+Kq), Vq=\{q-Kq),

we have therefore

S-OLl" 'Qt2n = i(<*l" * -«2n + «2n ' * 'Oil),

S-ai- • -C^n-l = i(tti* • 'tt2n~l — «2n-l' * 'Oil),

V'CXi- ' 'OL2n = !(«!« * ,Q;2n ~ «2n ' " '«t),

F'Qfi- • ■a2n-l = %(<Xi' ' -OL2n-\ + «2n-l ' * -«l).

* Consult Dickson: Linear Algebras, p. 11.

106 VECTOR CALCULUS

In particular

2Sa$ = aft + Pa, 2SaPy = afiy - y(3a,
2Vap = a/3 - fa 2Va(3y = a(3y + y(3a.

It should be noted that these formulae show us that both
the scalar and the vector parts of the product can them-
selves always be reduced to combinations of products.

This is simply a statement again of the fact that in
quaternions we have'only'one kind of multiplication, which
is distributive and associative.

We see from the expanded form above for S • qr that

S-qr = S-rq.

Hence, in any scalar part of a product, the factors may be
permuted cyclically. For instance,

S-afi = S-(3a, S-a(3y = S-Pya = S-yaQ,
S-a(3y5 = SPyfa

From the form of

Sq=Uq+Kq), Sq = SKq;

hence we have

Sa(3 = S@a, Safiy = - Syfa Sa(3y8 = S8y(3a, etc.

From the form of VKq = — Vq we see that

Vafi = - V@a, Vafiy = VyPa,
Vapyh m - Vdypa, Vapyhe = VebyPa-

We do not have a simple relation between V-qr and
V-rq, but we have the fact that they are respectively the
sum and the difference of two vectors, namely,

If a — SqVr -+- SrVq, P = VVqVr, then ft is perpendicular
to a, and

Vqr = a + P, Vrq = a — (3.

q = w +

w? + ,

;V + ^

(Tq)2 = w2 +

x2 +

2/2 + *2,

£g =

»,

(TTg)2 -

z2 +

f+z\

VECTORS IN SPACE 107

It is obvious that TVqr = TVrq and that /.qr = /rq
- tan-1 TVqr/Sqr. The planes differ.

The product of g and i£g is the square of the tensor of q.
We indicate the unitary part of q, called the versor of q,
by Uq. We have then the formulae

Kq = w — ix — jy — kz,
j j = w + ix + jy + kz
q Tq

Vq = ix + jy + kz,

TTVn _ix + jy + fa
^Kg rrg '

(TVUq)2 m (X* + f + *2)/(w2 + a? + 2/2 + z2),
cos- Z g = w/Tg = #• £7g,
sin-Z g = TVq/Tq= TVUq,
Z-q= tan"1 rFg/.Sg.

The product of two quaternions is

qr = ww* — xx' — yy' — zz' -f i(wx' -\- w'x + yzf — y'z)

+ j(wy' + w'y + zx' — z'x)

+ k(wz' + w'a + xyf — x'y).

From the formula Tqr = TgTY we have a noted identity

(ief +a*+ y2 + z2) <>'2+ x'2 +y'2 + s'2)
= (wwf — ao' — 2/2/' — zz')2 + (wa;' + w'x + 2/2' — S/'s)2
+ (wy' + to'y + zx' — z'x)2 + (W + w'z + #2/' — ^'2/)2-

This formula expresses the sum of four squares as the
product of the sums of four squares. It was first given by
Euler. The problem of expressing the sum of three squares
as the product of sums of three or four squares and the
sum of eight squares as the product of sums of eight squares
has also been considered.

108

VECTOR CALCULUS

9. Rotations. We see from the adjacent figure that we
have for the product

qrq-1

a quaternion of tensor and angle the same as that of r.
But the plane of the product is produced by rotating the
plane of r about the axis of q through an angle double the
angle of q. In case r is a vector /3 we have as the product
a vector fif which is to be found by rotating conically the
vector (3 about the axis of q through double the angle of q.
It is obvious that operators* of the type qQq~l, r()r-1,
which are called rotators, follow the same laws of multiplica-
tion as quaternions, since g(r()r_1)<7-1 = qrQ[qr]~l. A
gaussian operator is a rotator multiplied by a numerical
multiplier, and is called a mutation. The sum of two
mutations is not a mutation. As a simple case of rotator
we see that if q reduces to a vector a we have as the result
of after1 = /3' the vector which is the reflection of /3 in a.
The reflection of /3 in the plane normal to a is evidently

— a$orl.

EXAMPLES
(1) Successive reflection in two plane mirrors is equivalent
* QOq'1 represents a positive orthogonal substitution.

VECTORS IN SPACE 109

to a rotation about their line of intersection of double their
angle.

(2) Successive reflection in a series of mirrors all per-
pendicular to a common plane, 2h in number, making
angles in succession (exterior) of <pu, (P23, <&*••• is equivalent
to a rotation about the normal to the given plane to which
all are orthogonal, through an angle 6 = 2h — ir — 2(<p12
+ (pu + ••• + <P2h-i,2h) which is independent of the
alternate angles.

(3) Study the case of successive reflections in mirrors in
space at any angles.

(4) The types of crystals found in nature and possible
under the laws that are found to be true of crystals, are
solids such that every face may be produced from a single
given face, so far as the angles are concerned, by the
following op9rations :

I, the reversal of a vector, in quaternion

form — 1 .

A, rotation about an axis a an()oTn.

L4, rotatory inversion about a — an()a~u.

S, reflection in a plane normal to /5 — jSO/S-1 = /?()/?.

The 32 types of crystals are then generated by the succes-
sive combinations of these operations as follows:

Triclinic Ci Asymmetric 1.

d Centre-symmetric 1,-1.

Monoclinic Cs Equatorial 1, 0Q0.

d Digonal polar 1, a()a-1.

C2h Digonal equatorial 1, a()a;-1, a()a.

Orthorhombic C2v Didigonal polar 1, a()a~l, 0Q0, Sap = 0.

D2 Digonal holoaxial 1, a()a-\ fiQfi'1, Sap = 0.

Du Didigonal equatorial .... 1, a()a-1, POP'1, «()«,

SaP = 0,
A = al'20a-1'2.
Tetragonal d Tetragonal alternating . .1, — A.

Du Ditetragonal alternating. 1, — A, P{)P~X.

d Tetragonal polar 1, A.

110 VECTOR CALCULUS

Ctk Tetragonal equatorial. . .1, A, aQa.

C4* Ditetragonal polar 1, A, /3()/3.

D4 Tetragonal holoaxial .... 1, A, 0Q0~K

Dak Dietragonal equatorial . . 1, A, aQa, /3()/3_1.
Rhombohedral C8 Trigonal polar l,B, where B is a2l30<*~il3'

Czi Hexagonal alternating . .1, B, — B.

Ctv Ditrigonal polar 1, B, pQ0. •

D, Trigonal holoaxial 1, B, 0Q0T+.

Did Dihexagonal alternating . 1, B, j8()/8~l, 7O7, 7

bisects Z/3, B0.
Hexagonal Czh Trigonal equatorial 1,5, aQa.

Dzh Ditrigonal equatorial . . .1, B, aQa, jS()/3_1.

d Hexagonal polar 1, C, where C = a1/3()«~1/3.

dh Hexagonal equatorial . . . 1, C, aQa.

Civ Dihexagonal polar 1, C, /3()yS, where Sap = 0,

0 bisects angle of 7 and
Cy, Say = 0.

Di Hexagonal holoaxial .... 1, C, /3()/S_1.

Dan Dihexagonal equatorial. .1, C, a()a, pQ(3~l.

Regular T Tesseral polar ..1, aQa'1, PQP~X, Safi

= Spy = Sya = 0, L
where L = (a + fj

+ 7)0(«+/3 + 7)-1.

Th Tesseral central 1, aQa~\ 0Q/T1, 7O7"1,

L, aQa.
Td Ditesseral polar 1, aQa'1, 0Q0-\ 7O7"1,

L, (a + fi)Q(a + /3).
0 Tesseral holoaxial 1, aQa-\ 0Q0~lt yQy~l,

L, (a + p)Q(a + P)~K
Oh Ditesseral central 1, aQa~\ $00-*, yQy'1,

t,t{« + 0)Q(a+0?t

aQa.

The student should work out in each case the fuJl set of
operators and locate vectors to equivalent points in the
various faces.

Ref. — Hilton, Mathematical Crystallography, Chap. IV-
VIII.

(5) Spherical Astronomy. We have the following nota-
tion:

X is a unit vector along the polar axis of the earth,
h is the hour-angle of the meridian,

VECTORS IN SPACE 111

L = cos h/2 + X sin h/2,
i = unit vector to zenith,
j = unit vector to south,
k = unit vector to east, X = i sin I — j cos /, where I is

latitude,
li = unit vector to intersection of equator and meridian,

\x — i cos I -\- j sin I, aSX/x = SkX = Sk/j, = 0,
d = declination of star,

5 = unit vector to star on the meridian = X sin d + jjl cos d,
z = azimuth,
A = altitude.
At the hour-angle h, 8 becomes 8' = L~l8L.

The vertical plane through 8f cuts the horizon in
iVi8' = JSJ8' + kSk8', tan z = Sk8'/Sj8'.
At rising or setting z is found from the condition Sid' = 0.
The prime vertical circle is through i and k. The 6-hour
circle is through X and V\ji.
a — right ascension angle,
t = sidereal time in degrees,
h = t + a,

Lt = cos t/2 + X sin t/2,
La = cos a/2 + X sin a/2,
e = pole of ecliptic,

X = first point of aries = vernal equinox = Lrl^Lt}
s = longitude,
b = latitude,
M = cos s/2 + e sin s/2.

Problems. Given /, d, find A and z on 6-hour circle.
Sfx8' = 0.
/, d, find h and z on horizon.
/, d, find A.

I, d, A, find h and z, 8' = L~l8L = i
cos A + ? cos s + k sin 2.

112 VECTOR CALCULUS

/, d, h, find A and z.
a and d, find s and b.

(G) The laws of refraction of light from a medium of
index n into a medium of index n' are given by the equation

nVvct — n'Vva!

where v, a, a' are unit vectors along the normal, the
incident, and the refracted ray.
The student should show that

Investigate two successive refractions, particularly back
into the first medium.

(7) It is easy to show that if q and r are any two quater-
nions, and /3 = V • VqVr, we may write

(8) For any two quaternions

qiq'1 ± r_1) = (r =b q)f\ and = r(r ± q)~lq.

-± -

9 r

(9) If a, b, c are given quaternions we can find a quater-
nion q that will give three vectors when multiplied by a, b,
c resp. That is, we can find q, a, ft y such that

aq = a, bq = ft eg = 7. (R. Russell.)

We have a — — V • Vc/aVa/b, etc., or multiples of these.

(10) In a letter of Tait to Cayley, he gives the following:

(q+ r)()(g+ r)"1 = (qlr)xrQf-i(q/r)-*

= qiq-iryQiq-^-vq-1 = qh^Qq-^q-1,
(Vq+ Vr)()(Vq+ Fr)"1 = fa/rWJf^fo/r)-1/*,

VECTORS IN SPACE 113

where tan xA = a sin A/ (a cos A + 1), c sin 2la sin ra/3
+ cos 2la cos ra/3 = 2 (a cos o + & cos /S) V (6 sin |8),
2c + sin 2/a cos ra/3 = 2a sin a/ (6 sin /3).

Interpret these formulae.

10. Products of Several Quaternions. We will develop
some useful formulae from the preceding.

If we multiply a(3-(3a we have

a2(32 - S2a(3 - V2a(3.

Since Sax = 0, if x is a scalar,

&*/3t = SaVfry, Sa(3y8 = SaVfiyb, etc.
Since

2Va(3 = a(3 - (3a, 2Sa(3 = a(3 + 0ce,
ffiaV(3y = af3y — ay (3 — (3ya + 7/fa = 2(7/3o — 07/?)

= 2(y(3a + 7«/3 — ay {3 — yap).
For

2<S/?7 • a = /57a: + 7/fo = 2aSj3y = 0:187 + 0:7/?,
whence

0:187 — $70 = Yj8o — 07/?.
Therefore

VaV(3y = ySa(3 - (3Say.

Adding to each side ccSfiy, we have

Va(3y = aS(3y - (3Sya + ySa(3.
Since

]S = crtaft 0 = a^SaP + a~Wa$,

which resolves (3 along and perpendicular to a,

Sqrq-1 = Sr = qSrq-1,

Vqrq-1 = h^q~l - Kq~lKrKq)

= iC^a-1 — qKrq~l) = qVr-q~l.

That is, if we rotate the field, Sr and TTr are invariant.

114 VECTOR CALCULUS

Hence Vapy = VafiyaoT1 = aV(3ya-oTl and Vafty,
Vfiya. are in a plane with a and make equal angles with a.
For instance if a, /?, y, Vafly, Vfiya, Vyafi intersect a
sphere, then a, /?, y bisect the sides of the triangle Vafiy,
Vpya, Vya(3, a being opposite to Vya(3, etc. Evidently if
«i, (X2- • -an are n radii of a sphere forming a polygon, then
they bisect the sides of the polygon, given by Vaia2- • -an,
F«2«3- • '<xn, Vets- - -anaia2, • • -Van(xi- - -an-i. This ex-
plains the geometrical significance of these vectors. In
fact for any vector a and quaternion q, the vector a bisects
the angle between Vqa and Vaq, that is to say we construct
Vqa from the vector Vaq by reflecting it in a. The same
is true for any product, thus (3yde • • • vol is different from
a(3y8e • • • v only in the fact that its axis is the reflection in
a of the axis of the latter.

<M3 ' ' ' Qnqi differs from qiq2 • • • qn only in the fact that
its axis has been rotated negatively about the axis of q\
through double the angle of qi. Indeed

?2?3- • -qnqi = q~Kqiq2- ■ -qn)q\.

If we apply the formula for expanding VaVfiy to
V(Vafi)Vy8 = — V(Vy8)Va(3 we arrive at a most im-
portant identity:

V-VapVy8 = 8Sa$y - ySa$8

= - V-VydVafi = aS(3y8 - /3Say8.

From this equality we see that for any four vectors

8Sapy = aSfiyd + @Sya8 + ySaj38.

This formula enables us to expand any vector in terms of
any three non-coplanar vectors. Again
5Sapy - VpySad = V-aV(V(3y)8

= - V-aV8V$y = Fa(3Sy8 - VayS(38.

VECTORS IN SPACE 115

We have thus another important formula

SSofiy = Va(3Sy5 + VfiySaB + VyaS08,

enabling us to expand any vector in terms of the three
normals to the three planes determined by a set of three
vectors, that is, in terms of its normal projections. Since

aSPyS = VpySad + VytSefi + VbfiSay

and

(3Syda = Vay S{38 + VySSofi + VdaSPy,
we have

VVapVyd = VabSPy + VPySad - VayS(38 - VpbSay.

From this we have at once an expansion for Vafiyh, namely

Vctfyd = Va(3Sy8 - VaySpb + VabSPy

+ SapVyb - SayVpb + SabVpy.
Also easily

Sapyd = SaPSyd - SaySpd + SabSpy.
SVapVyb = SadSPy - SaySpb.
V-ap-Sybe = yS-VapVbe - bS-VapVye + eS- VapVyb

y b e

Say Sab Sae

SPy SPB Spe

In the figure the various points lie on a sphere of radius I.
The vectors from the center will be designated by the
corresponding Greek letters. The points X, Y, Z are the
midpoints of the sides of the A ABC. From the figure it
is evident that

H» = yli = (7/«1/2, v/y - «h m (a/7)1*,
Whence

7 = sar1, « - nrr\ p = ^r1,

116

VECTOR CALCULUS

P
Fig. 18.
where

v = it1!,

and the axis of p is ± a. Also p%p~l = ^iT1^7?-1* so that
if P is the pole of the great circle through XY then the
rotation pQp~l brings £ to the same position as the rotation
around OP through twice the angle of tjJ-1. Since £ goes
into {' by a rotation about OA as well as one about 0Pf
this means that the new position 0Zr is the reflection of OZ
in the plane of OP A. The angle of p is then ZAL or ZAP
according as the axis is -\- a or — a. The angles of L and
M are right angles, and if we draw CN perpendicular to
XY then

ANCY = ALAY, ANCX = AMBX,

and

AL = BM = CN and APB is isosceles.

Hence the equal exterior angles at A and B are ZAL
= ZBM = \{A + 5 + Q.

Draw PZ, then /ZiM = Zv^1 for it =JzJWM
= \ML = ZF since ilfZ - XN and iVF = YL. The
angle between the planes LAP and ZOP is thus the biradial
7)%~l and also £" is the biradial whose angle is that of the

VECTORS IN SPACE 117

planes OAZ, ZOP, so that ZOA and AOL make an angle
equal to z p, hence

ZV = h(A + B+C).
Further

pa'1 - nlyyfc'tla = («/t)1/2(t/«1/2(/3/«)1/2 - p'.

The angle of p' is thus %(A + 5 + C - tt) = 2/2 where S
is the spherical excess of AABC.

Consider the quaternion p = r)^1^ = — 77^". The con-
jugate of p is Kp = ££77, whose axis is also a and angle

- \{A + B + 0). Thus the quaternion ffij = - sin 2/2

- a: cos 2/2.

Shifting the notation to a more symmetric form we have
for any three vectors

aia2as = — sin 2/2 — TJVai(x2a.z • cos 2/2

= cos \<j — k sin Jo-,

where 2 is the spherical excess of the triangle the midpoints
of whose sides are A\, A2, A% and a is the sum of the angles
of the triangle. Hence

Saia2a3 = cos Jo", Va ia2a3 = ~* UV<x\ol2ccz sin \a.

It is to be noted that the order as written here is for a
positive or left-handed cycle from A\ to A2 and A$. Since
2 is the solid angle of the triangle, — S-a\a2as is the sine
of half the solid angle and — TVa\a2az is the cosine of half
the solid angle, made by oi, a2, a3.

If now we have several points as the middle points of the
sides of a spherical polygon, say aia2- • -an and the vertex
between a\ and an is taken as an origin for spherical arcs
drawn as diagonals to the vertices of the polygon, then for
the various successive triangles if we call the midpoints of
the successive diagonals

J*lj $2, ' ' "fn-3

118 VECTOR CALCULUS

we have, taking the axis to the origin which we will call k,
and which is the common axis of all the quaternions made
up by the products of three vectors

The sum of the angles of the polygon is the sum of the
angles of all the triangles into which it is divided, so that
if this sum is a we have for any spherical polygon

«i«2- • *«n = (— )n_3[cos cr/2 — k sin a/2].

We are able to say then that if the midpoints of the sides
of a spherical polygon are ai, a2, • • -ant then

SoCi(X2' ' '0in = db COS ff/2,

where a is the sum of the angles ; the vertices of the polygon
are given by

Wolioli- • -an, TJVcioOLz - • • anai, ••'•,
UVan- • -ttn-l,

each being the vertex whose sides contain the first and last
vectors in the product; and the tensors of these vectors are
each equal to sin <r/2.

The expression — Sa(3y is called the first staudtian of
afiy, the second staudtian is

- SVapVPyVya/TVapTVPyTVya

= S2aj3y/TVaPTV(3yTVya,

which is evidently the staudtian of the polar triangle.

S-ai--an ,iri i
mrz — — — • = tan f solid angle.
1 V •«!• • -an

We will summarize here the significance of the expressions
worked out thus far, and in particular the meaning of their
vanishing.

VECTORS IN SPACE 119

Sa(3 is the product of TaTp by the cosine of the angle
between a and — 0. It vanishes only if they are per-
pendicular.

Vafi is the vector at right angles to both a (3 whose length is
TaTfi multiplied by the sine of their angle. It vanishes
only if they are parallel.

Safiy is the volume of the parallelepiped of a fi y, taken
negatively. It vanishes only if they are all parallel to
one plane.

Vafiy, Vafiyd, • •'• these vectors are the edges of the poly-
hedral giving the circumscribed polygon, and if the ex-
pression vanishes, we have by separating the quaternion,

Va0y8- • • = aS(3y8- • • + VaVPyS-'- = 0.

Hence a is the axis of (3yd- • • and Sfiyd- • • equals zero.
By changing the vectors cyclically we have n vectors
all of which have a zero tensor, so that each edge is the
axis of the quaternion of the other n — 1 taken cyclically.
This quaternion in each case has a vanishing scalar.
n = 3, a j8 y are a trirectangular system.
n = 4, a (3 y 8 are coplanar, shown by the four vanish-
ing scalars. The angle a(3 = angle 7#.
n = 5, the edge Va(3y is parallel to V8e and cyclically

similar parallelisms hold.
We have in all these cases the sum of the angles of the
circumscribing polygon a multiple of 2w and it
satisfies the inequality S(n — 2)tt is greater than
a which is greater than {n — 2)x. It is evident
that if the polygon circumscribed has 540° the
vectors lie in one plane. ■
Safiyb = 0. If e = Va(3y8, then VaQySe = 0, and the

preceding case is at hand for the five vectors.
S-aia2- • -oLn = 0, the sum of the angles of the polygon is
an odd multiple of x.

120

VECTOR CALCULUS

EXERCISES

1. S-VaPVpyVya = - (Sapy)*
V-VapVpyVya = VaP(y*SaP - SPySya) + .....

2. S(a + P)iP + 7)(7 + «) m 2Sa0y.

3. 5-F(a + /3)(0 + 7)708 + 7)(7 + a)V(y + «)(a + 0)

4. 5.F(Fa/3F/37)(F/37^7«)7(F7aFa/3) = - (S-afiy)*.

5. S-5ef - - 16(5 -a^)4,
where

5 = F(F[« + 0[\fi + 7]F[^ + 7][7 + «]),
< = y(7D9 + 7][7 + a]V[y + a][a + fl),
f = V(V[y + «][a + /S]7[a + 0]\fi + 7]).

6. S(xa + yP + 27)(x'a + y'0 + *'7)(x"a + y"0 + *"7)

4(5.afl7)1.

7.

x \

X'
X"

S-aiPiyi =

-

Saai Sftai Syai
Sa0i S00i Syfii
Say 1 Sfiyi Syyi

Saai Sa&i

S0ai sm

Syai Syffi
S8ai S8P1

s
s
s

,8

ayi Sadi
Pyi SP81
771 Sy8i
571 S881

■■ 0

S • a/37.

for any eight vectors. If the element Saai is changed to Szai the value
is - S-0y8'S'Piyi8i-S-ai(e — a).

9. S-Va0yV0yaVyaP = ISaPSPySyaSaPy.

10. From S2P/a - V2Pfp = 1 we find

T(Sp/a + Vp/P)
where

1 = T{\cl+p + \pa~i

- irv + yr1) = T(a'P + p/80

a' = §(«T* - r>), pV = i(a"» + p*-«).

11. If TP = Ta = Tp = 1 and S-afip = 0,

S-U(p-a)U(p -P) = ±iV[2(l -Sap)].

12. If a, P, 7 and ah Pi, 71 are two sets of trirectangular unit vectors
such that if a = Py, ax = Piy, then we may find angles called Eulerian
angles such that

a2 = a COS yp + P sin \J/, P2 = — a sin i£ + P COS ^,

73 = 7 cos 6 + <*2 sin 0, a3 = — 7 sin 0 -f «2 cos 0,

71 = 73, «i = «3 cos ^ + /?2 sin <p,

Pi = — a3 sin v> + /32 cos <p.

VECTORS IN SPACE 121

13. If q = ai«2 • • • otn then if we reflect an arbitrary vector in
succession in a„, an-i, • • • 0:20:1 when Sq = 0 the final position will be a
simple reflection of p in a fixed vector, and if Vq = 0 the final position
will be on the line of p itself. Similar statements hold if the reflections
are in planes that are normal respectively to an, • • • «i.

11. Functions. We notice some expressions now of the
nature of functions of a quaternion. We have the follow-
ing identity which is useful :

(a/3)n + {$a)n = (ol$ + $a)l(<xP)n~l] ~ a^a[(a^n~2

= 2SaP[(a(3)n~1 + (/to)71-1] - a2^[(a^n~2

+ 08*)*-*].

Whence 2nSna(3 = (a/3 + Mn = [(«/3)n + (fax)"]

+ lt/ nl ni K«/5)"-2 + (/3a)""2]a2/32
\\{n — 1)1

+ 2l(nwl2)1 [(«»n_4 + w-v/34 + • • •

\\{n — 1)1

This implies the familiar formula for the expansion of cosn 0
in terms of cos nd, cos (n — 2)0, and we can write as the
reverse formula

S(a(3)n - (-)w/2[an/3n - n2S2a(3-an-2l3n-2l2\

+ n2(n2 - 22)SV-an"4/3"-4/4! - • • •] n even
(- ) (n~l) '2[nSa(3 • an~ler-lll !

- n(n2 - l2)53a/5-o:n-3/3n-3/3! + • • •] n odd.
Likewise

TV2na$= (-l)n/22n-1[S(al32n

(2n)!

l!(2n- 1)

S(aP2n~2a2p2 + ■••]

122 VECTOR CALCULUS

7»p»-ia/3== (_l)«/22«-2[7T(a/3)2n-1

- (2n - ^l TV(aB)2n~3 + • . .1
l!(2n-2)1 1VKfxp) x J

TV(ap)n/TVap = (-)n/2[n5aiS-Q:n-2/?n-2/l!

- n(n2 - 22)iS3a/3«n-4/Sn~4/3! + • • -J n even
(_1)<*-*^1 - (n2 - l^SPap-cT+p^fil + • • •] n odd.

Since jS/a is a quaternion whose powers have the same
axis we have (1 — 0/a)-1 = 1 + fi/a + 03/a:)2 + • • • when
Tfi < Ta, and taking the scalar gives the well-known
formula

Likewise

S-^~= 1 + S/5/a + S(/3/a)2 +
a — p

TV-^—= TVp/a + TV(p/a)2 +
a — p

If we define the logarithm as in theory of functions of a
complex variable we have

log (1 - fi/a) = log 7(1 - fi/a) + log 17(1 - fi/a)

= - fa - Itf/a)* - HP/a)* .

Therefore

log f(l - fi/a) - - Sfi/a - §S(/?/c*)2

Z °LZ_1 = TV log (1 - fi/a) = TVp/a + ^TV(p/a)2-
a

Again

T{a - p)~l = Ta'1 - f(l - P/a)-1 - fo^l +

Pi(- SUp/a) TP/a + P2(- SUp/a) T2P/a + .••],

where Pi P2 are the Legendrian polynomials.

Evidently for coaxial quaternions we have the whole
theory of functions of a complex variable applicable.

VECTORS IN SPACE 123

12. Solution of Some Simple Equations.

(1). If ap = a then p = oTla.

(2) . If Sap = a then we set Vap = f where £* is any vector
perpendicular to a, and adding, p = aa_1 + a~l$.

(3). If Fap = jS then *Sap = a: where # is any scalar, and
adding we have p = a~l(3 + aaaf"1.

(4). If Vapfi = y then SaVapQ = &x2p/3 = <*2£p/3 = Say
and SpVap(3 = /32£ap = S/fy. Now

Fap/5 = aS/3p - pSafi + (3Sap

and substituting we have

p = [o;-1^7 + /T1^ - y]/8afi.

The solution fails if Sa(3 = 0. In this case the solution is

p = _ a-'S^y - p^Sa-iy + xVofi,

x any scalar.

(5). If Yapp = 7 then Sa(3pSafi = &*07 and Soft)
= Sa(3y/Sa(3. Adding to Va(3p, we have

afip = 7 + Sa(3y/SaP and p = 0^or*7 + '(hcT*8cfiyl8c&.

(6). If &xp = a, £/3p = b, then a^p = zFa/3 + V(al3
- ba)Va(3.

(7). If Sap = a, S(3p = b, Syp = c, then

pSafiy = aV(3y + bVya + cFa/5.

(8). If gag-1 = |3 then g = (x/3 + y)/(a + /3) where x and
?/ are any scalars. Or we may write

q = u + 0(a + |8) + wFa?/3 where u = — w#a(a: + /3).

(9). If gag"1 = y, q^q'1 = 8, then

V(y - a)(8 - ft!

..

1 +

S(T + «)(«- ft

124 VECTOR CALCULUS

(10). If qaq-1 = f, qpq~l = *, qyq~l = f, then

S-flft - «) - 0, S-q(V - ft - 0, flf.gtf - 7) = 0,

hence Fg is coplanar with the parentheses, and we have

x(i - a) + 2/(77 - ft + H(f - 7) = 0
where

»:*:*->- 2S7(r? - ft : 2Sy(i - a) : S(£ + a)(i, - ft.

The six vectors are not independent. Vq is easily found
and thence Sq from

qa = £q.

(11). If (p - a)"1 + (p - ft"1 - (P ~ 7)"1 ~ (P ~ 5)-1
= 0, then if we let

ifi' ~ aT1 = 1 * (TO - 5)"1 - 5] - [(a - 6)"1 - 5])

= (p — 8)(p — a)_1(« — 5), etc.,

where p', a', 0', 7' are the vectors from D, the extremity of
5, to the inverses with respect to D, of the extremities of
p, a, ft 7, then

(p' - a')"1 + (p' - ft)"1 - (p' - 7T1 = 0.

Prove that

1 - ft _ y - ft _ P' - y _ r y - /ni/2

p

whence p' and p. (R. Russell.)

(12). If (q - a)"1 + (q - 6)"1 - (q - c)"1 - (q - d)~'
= 0, we set

(q - d)(q' - d)= (a- d){a' - d) = (b - d)(b' - d)

= (c - d){c' - d) - 1,

VECTORS IN SPACE 125

thence

(q - d)-> -(q- a)'1 = (4 - d)-\a - d)(q - a)'1
(q - d)~i [(a~d)/(q-a)+(b- d)l(q- b)- (c - d)/(q- e)]
- (?' - a')'1 + (?' - &T1 ~ W ~ cT1
and we have q' from

(V - cW - C) = (g' - 6')/(g' - «0

= (q' - c')l(a' - c') - [(V - c')Ka' - c')]K

(R. Russell.)

13. Characteristic Equation. If we write q = Sq + Vq
and square both sides we have q2 = S2q + (Vq)2 + 2Sq-Vq
whence

g2 - 2qSq + S2q - V2q = 0.

This equation is called the characteristic equation of q.
The coefficients

2Sq and S2q - V2q = T2q

are the invariants of q; they are the same, that is to say,
if q is subjected to the rotation r()r-1. They are also the
same if Kq is substituted for q. Hence they will not define
q but only any one of a class of quaternions which may be
derived from each other by the group of all rotations of the
form rQr~l or by taking the conjugate.
The equation has two roots in general,

Sq + Tqyl - 1 and Sq - Tq^ - 1.

Since these involve the V — 1 it leads us to the algebra of
biquaternions which we do not enter here, but a few re-
marks will be necessary to place the subject properly.

Since the invariants do not determine q we observe that
we must also have UVq in order to have the other two
parameters involved.

126 VECTOR CALCULUS

If we look upon UVq as known then we may write the
roots of the characteristic equation in the number field of
quaternions as Sq + TVqUVq and Sq — TVqUVq or

q and Kq.

If we set q -f- r for q and expand, afterwards drop all the
terms that arise from the identical equations of q and r
separately, we have left the characteristic equation of two
quaternions, which will reduce to the first form when they
are made to be equal. This equation is

qr+rq-2Sq-r- 2Sr-Vq + 2SqSr - 2SVqVr = 0.

We might indeed start with this equation and develop the
whole algebra from it.
We may write it

qr-\- rq- 2qSr - 2rSq + 4Sq-Sr + S-qr + S-rq = 0

which involves only the scalars of q, r, qr, and rq.

14. Biquaternions. We should notice that if the param-
eters involved in q can be imaginary or complex then
division is no longer unique in certain cases. Thus if

Q2=q2
we have as possible solutions

Q = ± q and also Q = ± V (- l)UVq-q.

If q2 = 0 and Vq = 0 then TVq = 0 and we have
Vq = x(i + j V — 1) where X is any scalar and i, j are any
two perpendicular unit vectors.

CHAPTER VII
APPLICATIONS

1. The Scalar of Two Vectors
1. Notations. The scalar of the product of two vectors
is defined independently by writers on vector algebra, as
a product. In such cases the definition is usually given for
the negative of the scalar since this is generally essentially
positive. A table of current notations is given. If a and (3
define two fields, we shall call S*cfi the virial of the two
fields.

S-a(3 = — a X /3 Grassman, Resal, Somoff, Peano, Bura-
li-Forti, Marcolongo, Timerding.

— Cfft Gibbs, Wilson, Jaumann, Jung, Fischer.

— a/3 Heaviside, Silberstein, Foppl, Ferraris,

Heun, Bucherer.

— (aft) Bucherer, Gans, Lorentz, Abraham,

Henrici.

— a|/3 Grassman, Jahnke, Fehr, Hyde.
Cos a/3 Macfarlane.

[a/3] Caspary.

For most of these authors, the scalar of two vectors,
though called a product, is really a function of the two
vectors which satisfies certain formal laws. While it is
evident that any one may arbitrarily choose to call any
function of one or more vectors their product, it does not
seem desirable to do so. For Gibbs, however, the scalar
is defined to be a function of the dyad of the two vectors,
which dyad is a real product. The dyad or dyadic of
Gibbs, as well as the vectors of most writers on vector
analysis, are not considered to be numbers or hypernumbers.

127

128 VECTOR CALCULUS

They are looked upon as geometric or physical entities,
from which by various modes of "combination" or de-
termination other geometric entities are found, called
products. The essence of the Hamiltonian point of view,
however, is the definition by means of geometric entities of
a system of hypernumbers subject to one mode of multiplica-
tion, which gives hypernumbers as products. Functions
of these products are considered when useful, but are called
functions.

2. Planes and Spheres. It is evident that the condition
for orthogonality will yield several useful equations, and
of these we will consider a few.

The plane through a point A, whose vector is a, per-
pendicular to a line whose direction is 8 has for its equation,
since p — a is any vector in the plane,

S-d(p-a) = 0.

If we set p = 8Sa/d we have the equation satisfied and as
this vector is parallel to 5 it is the perpendicular from the
origin to the plane. The perpendicular from a point B
is b~lS{a - 0)5.

If a sphere has center D and radius T(3 where /? and — (3
are the vectors from the center to the extremities of a
diameter, then the equation of the sphere is given by the
equation

S(p - 3 + fi)(p - d - P) = 0, orp2 - 2S8P + 52 - /32 = 0.

The plane through the intersection of the two spheres

p2 - 2£5ip + ci = 0 = p2 - 2S82P + c2

is 2S(5i — 52)p = ci — c2.

The form of this equation shows that it represents a plane

APPLICATIONS 129

perpendicular to the center line of the spheres. The point
where it crosses this line is

X18] + x282

P = i »

Xi + x2

whence solving, we find

p = v(h + 82)-\V8,82 + i(cj - <*)>.

3. Virial. If (3 is the representative of a force in direction
and magnitude then its projection on the direction a is
a~1Sa^f and perpendicular to this direction crWafi. If a
is in the line of action of the force, the projection is fit If a
is a direction not in the line of action then the projection
gives the component of the force in the direction a. If a
is the vector to the point of application of the force then
Sa(3 is the virial of the force with respect to a, a term intro-
duced by Clausius. It is the work that would be done by
the force in moving the point of application through the
vector distance a. If a fe an infinitesimal distance say,
8a, then — S8a(3 is the virtual work of a small virtual dis-
placement. The total virtual work would be 8V =
— 2S8an(3n for all the forces.

4. Circulation. In case a particle is in a vector field
(of force, or velocity, or otherwise) and it is subjected to
successive displacements 8p along an assigned path from
A to B, we may form the negative scalar of the vector
intensity of the field and the displacement. If the vector
intensity varies from point to point the displacements
must be infinitesimal. The sum of these products, if there
is a finite number, or the definite integral which is the limit
of the sum in the infinitesimal case, is of great importance.
If a point is moving with a velocity a [cm./sec] in a field of
force of /3 dynes, the activity of the field on the point is

130 VECTOR CALCULUS

— S-(3<t [ergs/sec.]. The field may move and the point
remain stationary, in which case the activity is S-(3a. The
activity is also called the effect, and the power. If <r is the
vector function of p which gives the field at the point P we
have for the sum

- 2Sa8p or - // Sa8p.

This integral or sum is called the circulation of the path for
the field a.

5. Volts, Gilberts. For a force field the circulation is the
work done in passing from A to B. If the field is an electric
field E, the circulation is the difference in voltage between
A and B. If the field is a magnetic field H, then the circula-
tion is the difference in gilbertage from A to B. It is
measured in gilberts, the unit of magnetic field being a
gilbert per centimeter. There is no name yet approved for
the unit of the electrostatic field, and we must call it volt
per centimeter. The unit of force is the dyne and of work
the erg.

6. Gausses and Lines. In case the field is a field of flux
a, and the vector TJv is the outward normal of a surface
through which the flux passes, then

- SaUv

is the intensity of flux normal to or through the surface
per square centimeter. The unit of magnetostatic flux B
is called a gauss; the unit of electrostatic flux D is called a
line. The total flux through a finite surface is the areal
integral

— fSaUvdA, written also — fSadv.

The flux-integral is called the transport or the discharge.
Thus if D is the electric induction or displacement, the

APPLICATIONS 131

discharge through a surface A is — fSDUvdA, measured
in coulombs. Similarly for the magnetic induction B,
the discharge is measured in maxwells.

7. Energy-Density. Activity-Density. Among other
scalar products of importance we find the following. If
E and D are the electric intensity in volts/cm. and induction
in lines at a point, — |$ED is the energy-density in the
field at the point in joules/cc. If H and B, likewise, are the
magnetic intensity in gilberts/cm., and gausses, respectively,
— 2^#HB is the energy in ergs. If J is the electric cur-
rent-density in amperes/cm.2, — S • E J is the activity in
watts/cc. If G is the magnetic current-density in heavi-
sides*/cm.2, — S ■ H G is the activity in ergs/sec. If the
field varies also, the electric activity is — >S- E(J + D) and
the magnetic activity — $H(G + B).

EXERCISES

1. An insect has to crawl up the inside of a hemispherical bowl, the
coefficient of friction being 1/3, how high can it get?

2. The force of gravity may be expressed in the form a = — mgk.
Show that the circulation from A to B is the product of the weight by
the vertical difference of level of A and B.

3. If the force of attraction of the earth is <r = — hUp/p2 show
that the work done in going from A to B is

hiTa-1 - T0-1].

4. The magnetic field at a distance a from the central axis of an
infinite straight wire carrying a current of electricity of / amperes is

H = 0.2ia-1(— sin di + cos 6j) (i andj perpendicular to wire)
and the differential tangent to a circle of radius a is ( — a sin 6 i
+ a cos 9j)dd. Show that the gilbertage is 0.2/ (02 — 0i) gilberts,
which for one turn is OAirl.

Prove that we get the same result for a square path.

5. The permittivity k of a specimen of petroleum is 2 [abfarad/cm.],
and on a small sphere is a charge of 0.0001 coulomb. The value of
the displacement D at the point p is then

D = 9^2 UplTp2 [lineg]

* A heaviside is a magnetic current of 1 maxwell per second.

132 VECTOR CALCULUS

What is the discharge through an equilateral triangle whose corners
are each 4 cm. from the origin, the plane of the triangle perpendicular
to the field?

6. If magnetic inductivity p. is 1760 [henry/cm.] and a magnetic
field is given by

H = la [gilbert/cm.],

then the magnetic induction is

B = 7 -1760a [gausses].

What is the flux through a circular loop of radius a crossing the field
at an angle of 30°?

7. If the velocity of a stream is given by

<r = 24(cos 6 i -f sin dj),

what is the discharge per second through a portion of the plane whose
equation is Sip = — 12 from

d = 10° to 6 = 20°?

8. The electric induction due to a charge at the origin of e coulombs is

D = - eUp/TPHir [lines].

What is the total flux of induction through a parallelepiped whose
center is the origin?

9. The magnetic induction due to a magnetic point of m maxwells is

B = - mUp/Tp2 [gausses].

What is the total flux of induction through a sphere whose center is
the point?

10. In problem 8, if the permittivity is 2 = k, then the electric
intensity

E = rH>4r.

What is the amount of energy enclosed in a sphere of radius 3 cm. and
center at a distance from the origin of 10 cm.?

11. In problem 9, if the inductivity is 1760 and the magnetic in-
tensity is

H = p~%

how much energy is enclosed in a box 2 cm. each way, whose center
is 10 cm. from the point and one face perpendicular to the line joining
the point and the center?

12. If the current in a wire 1 mm. in diameter is 10 amperes and
the drop in voltage is 0.001 per cm., what is the activity?

APPLICATIONS 133

13. If there is a leakage of 10 heavisides through a magnetic area of
4 cm.2, and the magnetic field is 5 gilberts/cm., what is the activity?

14. Through a circular spot in the bottom of a tank which is kept
level full of water there is a leakage of 100 cc. per second, the spot
having an area of 20 cm.2. If the only force acting is gravity what is
the activity?

15. If an electric wave front from the sun has in its plane surface
an electric intensity of 10 volts per cm., and a magnetic intensity of
0033 gilberts per cm., and if for the free ether or for air y. = 1 and
k = £-10~20, what is the energy per cc. at the wave front? (The
average energy is half this maximum energy and is according to Langley
4.3 -10-5 ergs per cc. per sec.)

16. If a charge of e coulombs is at a point A and a magnetic point
at B has m maxwells, what is the energy per cc. at P, any point in space,
the medium being air?

8. Geometric Loci in Scalar Equations.
(1). The equation of the sphere may be written in each
of the forms

a/p = Kp[a,

S(p - a)/(p + a) = 0,

S2a/(p + a) = 1,

S2p/(p + <*) - 1,

T(Sp/a + Vp/a) = 1,

Tip - ca) m T(cp - a),
S{p - a) (a - »08 - 7)(Y - B)(S - p) - 0,

a2Sfiyp + j32Syap + y2Sa(3p = p2Sa(3y
0 (p-aO2 (p-/3)2 (p-7)2 (P-5)2

(p - a)2 0 (a - /3)2 (a - y)2 (a - 5)2

(p-/?)2 (/? -«)2 0 (/5-t)2 (0-S)2

(P-T)2 (Y-«)2 (Y-0)2 0 (7-5)

(p - 5)2 (5 - a)2 (5 - /3)2 (5 - 7)2 0

Interpret each form.

(2). The equation of the ellipsoid may be written in the
forms

S2p/a - V2p/(3 = 1,

where a is not parallel to ft

T(p/y + Kpjb) = T(p/8 + tfp/7),

rOup + pX)=x2-/*2.

134 VECTOR CALCULUS

The planes

a p

cut the ellipsoid in circular sections on Tp = Tfi. These
are the cyclic planes. Tfi is the mean semi-axis, Ufi the
axis of the cylinder of revolution circumscribing the ellip-
soid, a is normal to the plane of the ellipse of contact of
the cylinder and the ellipsoid.
In the second form let

r1 - -£, 7-1 = - £> t2 = n2 - TJ,

then the semi-axes are

a=rX+7>, 6= ^~ TfX*> c=T\-T».
T(\ - n)

(3). The hyperboloid of two sheets is S2p/a + F2p//3 = 1.
(4). The hyperboloid of one sheet is S2p/a + V2p/(3 = — 1.
(5) . The elliptic paraboloid of revolution is

SplP+V2p/(3 = 0.

(6). The elliptic paraboloid is Sp/a + V2p/(3 = 0.
(7). The hyperbolic paraboloid is Sp/a Sp/fi = Sp/y.
(8). The torus is

T(± bUarWap - p) = a,

2bTVap = ± (Tp2 +b2- a2),

4b2S2ap = 4b2T2p - (T2p + b2 - a2)2,

Aa2T2p - 4b2S2ap = (T2p - b2 + a2)2,

SU(p - «V (a2 - b2))l(p + cW (a2 - b2)) = ± b/a,

p = ± bJJoTWar + at/Y, r any vector.

(9). Any surface is given by

p = <p(u, v).

APPLICATIONS 135

A developable is given by p = <p(t) + ucp'it).
(10). A cone is f(U[p - a]) = 0.
The quadric cone is SapSfip — p2 = 0.
The cone through a, (3, y, 8, e is

S-V(Va(3V8e)V(V(3yVep)V(Vy8Vpa) = 0,

which is Pascal's theorem on conies.

The cones of revolution through X, n, v are

The cones of revolution which touch S\p = 0, Sfxp = 0,
Svp = 0, are

The cone tangent to (p — a)2 + c2 = 0 from /? is

c2(p -a-$)2= V2(3(p - a).

The polar plane of /3 is £/3(p — a) — — c2.
The cone tangent to

a p

from 7 is

(*i-F,S-1)(fl,J'-pi-0

-( S^S^-- SV?V?-- lY=0.
\ a a a a /

The cylinder with elements parallel to y is

(s*i-f1-i)H-p?)

_(s>sl-sv>vl)2 = o.

\ a a a a)

136 VECTOR CALCULUS

For further examples consult Joly : Manual of Quater-
nions.

2. The Vector of Two Vectors
Notations, If a and /3 are two fields, we shall call V-a(3
the torque of the two fields.

Va(3 = Va(3 Hamilton, Tait, Joly, Heaviside, Foppl,
Ferraris, Carvallo.
cqS Grassman, Jahnke, Fehr.

aX 0 Gibbs, Wilson, Fischer, Jaumann, Jung.
[a, /3] Lorentz, Gans, Bucherer, Abraham, Timer-
ding.
[a | /?] Caspary .

a A j3 Burali-Forti, Marcolongo, Jung.
aj8 Heun.

Sin a/3 Macfarlane.

Iaccb Peano.
1. Lines. The condition that two lines be parallel is that
Vafi = 0. Therefore the equation of the line through the
origin in the direction a is Vap — 0.

The line through 0 parallel to a is Va(p — fi) = 0 or
Vap = Va(3 = y. The perpendicular from 5 on the line
Vap = 7 is

— a~lVab + a~ly.

The line of intersection of the planes, S\p = a, S^p = b, is
VpV\fx = a/x — 6X. If we have lines Vpa — y and Vp& = 8
then a vector from a point on the first to a point on the
second is 5/3"1 — 7a-1 + #/3 — ya. If now the lines in-
tersect then we can choose x and y so that this vector will
vanish, corresponding to the two coincident points, and
thus

S{bp~l - ya~l)$a = 0 = S8a + Syp.

APPLICATIONS 137

If we resolve the vector joining the two points parallel and
perpendicular to Vaft we have*

5/3-1 — ya~l + xfi — ya

= • (Va^S • VaP(bprl - yoT1 + zp - ya)

= -(VaP)-\S5(x+ Spy)

L a Fa/3 Fa/3 J

L Va0 P Vap]

- «-* f- SaPS ^~ + a2S JL 1
Vap Vap]

Hence the vector perpendicular from the first line to the
second is

- (Vafl-KStct + Spy)

and vectors to the intersections of this perpendicular with
the first and second lines are respectively

and

ya x — a 1 \ 8 ' — ^—

L Va(3 J

* Note that

(Va0)-lV(Vu0)(z0 - ya) = xp - ya
(y«jS)-1F-7a/3(5J3-1 - ya~l) = (Vc0)-l(- a'^Sfiya - (rlS<*&)

Va,(-^S^+p-S^)

10

138 VECTOR CALCULUS

The projections of the vectors a, y on any three rectangular
axes give the Pluecker coordinates of the line. For applica-
tions to linear complexes, etc., see Joly: Manual, p. 40,
Guiot: Le Calcul Vectoriel et ses applications.

2. Congruence. The differential equation of a curve or
set of curves forming a congruence whose tangents have
given directions cr, that is, the vector lines of a vector field <r,
is given by

Vdpa = 0

or its equivalent equation

dp = adt.

3. Moment. The moment of the force /3 with respect
to a point whose vector from an origin on the line of @ is
a, is — Fa/3. If the point is the origin and the vector to
some point in the line of application of the force is a, then
the moment with respect to the origin is Vafi. If the
point is on the line of application the moment obviously
vanishes. If several forces have a common plane then
the moments as to a point in the plane will have a common
unit vector, the normal to the plane. If several forces are
normal to the same plane, their points of application in
the plane given by ft, ft, ft, • • • , their values being a\a>
a2a, asa, • • • , then the moments are

F(aift + a2ft + a3ft + • • •)« [dyne cm.].
If we set

«ift + 02ft + 03ft + • • • = ftai + a2+ az+ • • •)/

then /3 is the vector to the mean point of application, which,
in case the forces are the attractions of the earth upon a
set of weighted points, is called the center of gravity. If
ai + #2 + a3 + • • • = 0, we cannot make this substitution.

APPLICATIONS 139

4. Couple. A couple consists of two forces of equal
magnitude, opposite directions and different lines of action.
In such case the mean point becomes illusory and the sum
of the moments for any point from which vectors to points
on the lines of action of the forces are ah a2 respectively, is

V{ax - a2)P.

But a\ — a2 is a vector from one line of action to the other,
and this sum of the moments is called the moment of the
couple. It is evidently unchanged if the tensor of /? is
increased and that of a\ — a2 decreased in the same ratio,
or vice versa.

5. Moment of Momentum. If the velocity of a moving
mass m is a cm./sec, then the momentum of the mass is
defined to be ma gr. cm./sec. The vector to the mass
being p, the moment of momentum of the mass is defined
to be

Vpma = mVpa [gm. cm.2/sec.].

6. Electric Intensity. If a medium is moving in a mag-
netic field of density B gausses, with a velocity a cm./sec,
then there will be set up in the medium an electromotive
intensity E of value

E=Fo-B-10~8 [volts/centimeter].

For any path the volts will be

- fSdPE= + fSdpBa-10-8.

If this be integrated around any complete circuit we shall
arrive at the difference in electromotive force at the ends
of the circuit.

7. Magnetic Intensity. If a magnetic medium is moving
in an induction field of D lines, with a velocity a, then there
will be produced in the medium at every point a magnetic

140 VECTOR CALCULUS

intensity field

H = OAwVDa [gilberts/cm.].

For any path the gilbertage will be OAirf SdpaD.

8. Moving Electric Field. If an electric field of induc-
tion, of value D lines, is moving with a velocity a, then
there will be produced in the medium at the point a mag-
netic field of intensity H gilberts/cm. where

H m OAirVaD.

For a moving electron with charge e, this will be
— (eUp/4:irTp2). For a continuous stream of electrons
along a path we would have

the point being the origin.

9. Moving Magnetic Field. If a magnetic field of in-
duction of value B gausses is moving with a velocity cr,
it will produce at any given point in space an electric
intensity E = V - BolO-8 volts per centimeter.

10. Torque. If a particle of length dp is in a field of
intensity <r which tends to turn the particle along the lines
of force, then the torque produced by the field upon the
element is

V-dpa.

If a line runs from A to B, the total torque is

// V-dpe.
For instance if dp, or in case of a non-uniform distribution
cdp, is the strength in magnetic units, maxwells, of a wire
magnet from A to B, in a field a, then

fIV-dpa or f/V-cdpa

is the torque of the field upon the magnet.

APPLICATIONS 141

11. Poynting Vector. An electric intensity E volts/cm.
and magnetic intensity H gilberts/cm. at a point in space
are accompanied by a flux of energy per cm.2 R, given by
the formula

4xR = — — [ergs/cm.2 sec.].

This is the Poynting vector.

12. Force Density. The force density in dynes/cc. of
a field of electric induction on a magnetic current is given by,

F = 4ttFDG : 10 [dynes/cc],

where D is the density in lines of electric displacement
G is the magnetic current density in heavisides per cm.2.
If the negative of F is considered we have the force per cc.
required to hold a magnetic current in an electrostatic
field of density D.

The force density in dynes/cc. of a field of magnetic
induction on a conductor carrying an electric current is

F-ijr.il.

A single moving charge e with velocity a will give
F =AweVaiJiVaD.

13. Momentum of Field. The field momentum at a
point where the electric induction is D lines and magnetic
induction B gausses is T = 3-109 V- DB [gm. cm./sec.]. If
the magnetic induction is due to a moving electric field then
T = 0.047rF- D/jlVDct, and if the electric induction is due
to a moving magnetic field,

T = VB/cVaB.

47T-3-1010

142 vector calculus

3. The Scalar of Three Vectors

1. Area and Pressure. If we consider two differential
vectors from the point P, say dip, d2p, then the vector area
of the parallelogram they form is Vdipd2p. If then we have
a distribution of an areal character, such as pressure per
square centimeter, /3, the pressure normal to the differential
area will be in magnitude

— S(3dipd2p.

The vector Vdipd2p may be represented by dp or JJvdA.
The vector pressure normal to the surface will be

UpS(3dipd2p.

There will also be a tangential pressure or shear, which is
the other component of /3.

2. Flux. If j8 is any vector distribution the expression
— S($d\pd2p is often called the flux of /? through the area
Vd\pd2p. It is to be noted however that the dimensions
of the result in physical units must be carefully considered.
Thus the flux of magnetic intensity is of dimensions that
do not correspond to any magnetic quantity.

3. Flow. If /3 is the velocity of a fluid in cm./sec, then
the volume passing through the differential area per second
is

— Sfid\pd2p [cc./sec.].

4. Energy Flux. The dimensions of the Poynting energy
flow R show that it is the current of energy per second across
a cm.2, hence the total flow per second through an area is

-SRd^p--8-™™^ [ergs/sec]

In the case of a straight conductor carrying a current of
electricity, we have at a distance a from the wire in a

APPLICATIONS 143

direction at right angles to the wire directly away from it
the value

T R= (4ir)-1108JS;(0.2Ja-1).

Consequently if we consider one centimeter of wire in
length and the circumference of the circle of radius a we
shall have a flux of energy for the centimeter equal to

J(ft-«0 [jouks].

This is the usual J2R of a wire and is represented by heat.

5. Activity. For a moving conductor we have already
expressed the vector E, and as the current density J can
be computed from the intensity of the field (J = k E) we
have then for the expression of the activity in watts per
cubic centimeter of conductor

A= -SaBhO-8= -S(V(rB)k(VaB)-lO-ie [watts].

Likewise in the case of the magnetomotive force due to
motion and the magnetic current G = IH we have for the
activity per cubic centimeter of circuit

A=- SDaQ = - S-(VDa)l(VD(r)-10-7 [watts].

6. Volts. The total electromotive difference between
two points in a conductor is the line-integral along the
conductor

- fSdpaBlCr* [volts].

7. Gilberts. The total magnetomotive difference be-
tween two points along a certain path is the line-integral

— AirfSdpDo- [gilberts].

4. Vector of Three Vectors
1. Stress. We find with no difficulty the equations

V-a(Ua± Uy)y = ± TyTa(Ua± Uy)
and

V-a(Vay)y = — Say -V -ay.

144 VECTOR CALCULUS

If now we have a state of stress in a medium, given by its
three principal stresses in the form

0i = g — 7V dynes/cm.2 normal to the plane orthogonal

to U(U\+ Un),
92 = g — S\fx dynes/cm.2 normal to the plane orthogonal

to UV\fi,
gz = g + T\p dynes/cm.2 normal to the plane orthogonal
to U{U\ - Un),

gi < gi < gz,

then the stress across the plane normal to /? is

V\fo + 0.

If the scalars git g%, gz are dielectric constants in three
directions (trirectangular) properly chosen, then the dis-
placement is

D = FXE/x + gE.

If the scalars are magnetic permeability constants,

B = V\Hfi + gW.
If the scalars are coefficients of dilatation, then 0 becomes

(T-- VWp+gp.

If the scalars are elasticity constants of the ether, then
according to Fresnel's theory, the force on the ether is,
for the ether displacement ft . .

V\fo + gp.

If the scalars are thermoelectric constants in a crystal,
then

D = FXQm + gQ. where Q is the flow of heat.

If g = 0 the scalars are TV, - TV, - SXfi. If V\fi = 0,
the scalars are 7V> — TV> T\p, that is, practically — t
along X and + t in all directions perpendicular to X.

CHAPTER VIII

DIFFERENTIALS AND INTEGRALS
1. DlFFEKENTIATION AS TO A SCALAR PARAMETER

1. Differential of p. If the vector p depends upon the
scalar parameter t, say

p = <p(t),

then for two values of t which are supposed to be in the
range of possible values for t

Pi — Pi = <p(h) — <p(ti) t
ti — t\ t% — ^1

If now we suppose that U < h < t2 and that h and t2 can
independently approach the limit, t0, then we shall call
the limit of the fraction above, if there be such a limit, the
right-hand derivative of p as to t, at t0, and if t2 < h < t0,
we shall call the limit the left-hand derivative of p as to
t at t0. In case these both exist and are equal, and if p
has a value for t0 which is the limit of the two values of
<p(ti), then we shall say that p is a continuous function of
t at t0 and has a derivative as to t at to.

There is no essential difference analytically between the
function <p and the ordinary functions of a single real
variable, and we will assume the ordinary theory as known.

It is evident that for different values of t we may con-
sider the locus of P which will be a continuous curve.
Since p2 — pi is a chord of the curve the limit above will give
a vector along the tangent of the curve. Further the tensor
of the derivative, Tp' = T(p'{t)y is the derivative of the
length of the arc as to the parameter t. If the arc s is the
parameter then the vector p' is a unit vector.

145

146 VECTOR CALCULUS

EXAMPLES

(1) The circle

p = a cos 0 -f 0 sin 0, To: = Fft Sa0 = 0,

p' = — a sin 0 + 0 cos 0.

(2) The helix

p = a cos 0 + 0 sin 0 + 70,
p' = — a sin 0 + 0 cos 0+7.

(3) The conic

_ a*2 + 20< + 7
P at2 +2U+C '

Multiplying out, t2(a - ap) + 2*(0 - bp) + (7 - cp) = 0
for all values of £. For 2 = 0, p = 7/c, and for t = 00 ,
p = a/a, hence the curve goes through a/a and 7/c.
We have

rfp/d< = [t2(ba — a0) + <(ca — 07) + (c0 — by)] times scalar.

Hence for t = 0, the direction of the tangent is 0/6 — 7/c
at 7/c, for t = 00, the direction of the tangent is
0/6 — a/a at a/a. Since these vectors both run from the
points of tangency to the point 0/6, the curve is a conic,
tangent to the lines through 0/6 and the two points a/a
and 7/c, at these two points. If the origin is taken at
0/6, so that p = w + 0/6, and if a! = a/a — 0/6, 7' = 7/c
- 0/6, then

at\a! - tt) - 26/tt + c(yf - w) = 0

is the equation of the curve.

If now we let w run along the diagonal of the parallelo-
gram whose two sides are a'y' so that tt = x(a! + y'), then
substituting we have

at2x + 2btx - c(l - x) = 0,

at2(l - x) - 2btx - ex = 0.

DIFFERENTIALS 147

From these equations we have

t2 = c/a
and

x = Vac/2(Vac± b).

These values of x give us the two points in which the
diagonal in question cuts the curve. The middle point
between these two is

Referred to the original origin this gives for the center

, ,, ca - 2b(3 + ay

k= r + p b = — — =£— •

2(ac — b2)

If we calculate the point on the curve for

bh + e

ah+ b

we shall find that for the points p2, pi we have J(p2 + Pi)
= k, so that k is the center of the curve and diametrically
opposite points have parameters

h and t2 = — — r-x >
ati ~t o

an involutory substitution. If ac = b2, k becomes co ex-
cept when also the numerator = 0. [Joly, Manual, Chap.
VII, art. 48.]

In general the equation of the tangent of any curve is

IT = p + Xp'.

We may also find the derivatives of functions of p, when
p = (p(t), by substituting the value of p in the expression
and differentiating as before. Thus

let p = a cos 6 + P sin 6 where Ta # Tp.

148 VECTOR CALCULUS

Then

Tp = V [- a2 cos2 6 - 2Sap sin 6 cos 6 - 02 sin2 6],

We may then find the stationary values of Tp in the manner
usual for any function. Thus differentiating after squaring

a2 sin 26 - 2Sa(3 cos 26 - fi2 sin 26 = 0,
tan 26 - 2Sap/(a2 - /32).

2. Frenet-Serret Formulae. Since the arc is essentially
the natural parameter of a curve we will suppose now that
p is expressed in terms of s, and accents will mean only
differentiation as to s. Then both

p and p + dsp'

are points upon the curve.

The derivative of the latter gives p' + dsp", which is also
a unit vector since the parameter is s. Thus the change in
a unit vector along the tangent is dsp", and since this
vector is a chord of a unit circle its limiting direction is
perpendicular to p', and its quotient by ds has a length whose
limit is the rate of change of the angle in the osculating plane
of the tangent and a fixed direction in that plane which
turns with the plane. That is to say, p" in direction is
along the principal normal of the curve on the concave side,
and in magnitude is the curmture of the curve, which we
shall indicate by the notation

Unit tangent is a = p',

Unit normal is |9 = Up", curvature is Ci = Tp",

Unit binormal is y = Va(3, so that Ciy = Vp'p".

The rate of angular turn of the osculating plane per centi-
meter of arc is found by differentiating the unit normal of
the plane. Thus we have

Ti = cf2hW - Fp'p"-c2].'

DIFFERENTIALS 149

But d2 = T2p" = - Sp"p" and therefore Clc2 = - SP"pf".
Substituting for c2 we have

71 - cr3[- Sp"p"Vp'pf" + SpV'Wl
= crz[Vp,Vp"Vp,,,p"]

= crWaVc1(3Vp'"c1p

= crW-aPVp"^ = crlVyVp'"p = crlpSyp"'

= - «lft

where «i is written for the negative tensor of 71 and is the
tortuosity. It may also be written in the form

Again since /? = ya we have at once the relations

j3i = 7i« + 7«i = «i7 ~~ C\a.

Thus we have proved Frenet's formulae for any curve

«i = erf, ft = ai7 — ci«, 71 = — a$.

It is obvious now that we may express derivatives of any
order in terms of a, ft y, and Oi, Ci, and the derivatives of
ai and Ci.

For instance we have

Pi = OL, p2 = fci,

Pa = ftci + Pc2 = fe + (701 — aci)ci,
Pi = 0c3 + 2{yai — aci)c2 + (ya2 — ac2)ci

- ^(ai2 + Cl2)Cl.

The vector w = aai + 7C1 is useful, for if 77 represents in
turn each one of the vectors a, /3, 7, then 771 = Fa^ It is
the vector along the rectifying line through the point.
The centre of absolute curvature k is given by

K = p - lip" m p + Pld.

150 VECTOR CALCULUS

The centre of spherical curvature is given by

a = k + yd/da • c{~1 = k — yc2/aiCi2.

The polar line is the line through K in the direction of 7.
It is the ultimate intersection of the normal planes.

3. Developable s. If we desire to study certain de-
velopables belonging to the curve, a developable being the
locus of intersections of a succession of planes, we proceed
thus. The equation of a plane being S(w — p)rj = 0,
where t is the vector to a variable point of the plane, and
p is a point on the curve, while rj is any vector belonging
to the curve, then the consecutive plane is

S(t - p)f) + ds'd/dsS(w - p)r) = 0.

The intersection of this and the preceding plane is the line
whose equation is

7r = p + (— r)Sar) + t)lVr}r}i.

This line lies wholly upon the developable. If we find a
secOnd consecutive plane the intersection of all three is a
point upon the cuspidal edge of the developable, which is
also the locus of tangents of the cuspidal edge. This vector
is
tv = p + (VwySar} + 2Vr)7]iSar)i + Vr}7}iS^rjCi)/ST]r}ir]2'

By substituting respectively for 77, a, ft 7, we arrive at the
polar developable, the rectifying developable, the tangent-
line developable.

EXAMPLE
Perform the substitutions mentioned.

4. Trajectories. If a curve be looked upon as the path
of a moving point, that is, as a trajectory, then the param-
eter becomes the time. In this case we find that (if
p = dp/dt, etc.) the velocity is p = av, the acceleration is

DIFFERENTIALS 151

p = ficiv2 + av. The first term is the acceleration normal
to the curve, the centrifugal force, the second term is the
tangential acceleration. In case a particle is forced to
describe a curve, the pressure upon the curve is given by
(3civ2. There will be a second acceleration, p = a(v — wi2)
+ (3(2cii + c2v) + yaiCiV. The last term represents a
tendency per gram to draw the particle out of the osculating
plane, that is, to rotate the plane of the orbit.

5. Expansion for p. If we take a point on the curve
as origin, we may express p in the form

p = sa + %cis2(3 — %s*(ci2a — c2/3 — cmy)

— ^4(3c2cia: ~~ £IC3 ~~ c* — Clttl2l ~~ T[2c2ai + da2])

EXERCISES

1. Every curve whose two curvatures are always in a constant ratio
is a cylindrical helix.

2. The straight line is the only real curve of zero curvature every-
where.

3. If the principal normals of a curve are everywhere parallel to a
fixed plane it is a cylindrical helix.

4. The curve for which

Ci = 1/ms, ai = 1/ns,
is a helix on a circular cone, which cuts the elements of the cone under
a constant angle.

5. The principal normal to a curve is normal to the locus of the
centers of curvature at points where Ci is a maximum or minimum.

6. Show that if a curve lies upon a sphere, then

cr1 = A cos a + B sin a = C cos (a + e), A, B, C, e are constants.
The converse is also true.

7. The binormals of a curve do not generate the tangent surface of
a curve.

8. Find the conditions that the unit vectors of the moving trihedral
afiy of a given curve remain at fixed angles to the unit vectors of the
moving trihedral of another given curve.

Two Parameters
6. Surfaces. If the variable vector p depends upon two
arbitrary parameters it will terminate upon a surface of

152 VECTOR CALCULUS

some kind. For instance if p = <p(u, v), then we may
write for the total differential of p

dp = dud/du(<p) -f- dvd/dv((p) = du<pu + dv<pv.

We find then

Fdp = £dw2 + 2Fdudv + GW,
where

E = — ^tt2, F = — S<pu<pv, G = — ^t,2.

We have thus two differentials of p, one for » = constant,
one for u = constant, which will be tangent to the para-
metric curves upon the surface given by these equations,
and may be designated by

pidu, p2dv.

The normal becomes then

v = vPlp2, Tv = V (EG - F2) = H.

For certain points or lines v may become indeterminate,
the points or lines being then singular points or singular
lines.

7. Curvatures. If we consider the point p and the
point p + dupi -f- dvp2 the two normals will be

v and v + duV(pnp2 + P1P12)

-f- dvV(pi2p2 + P1P22) + • • •
which may be written

v and v + dv.

The equations of these lines are

V(w - P)v =0, V(w- p- dP)(v + dv) = 0.

They intersect if

Sdpvdv = 0.

Points for which this equation holds lie upon a line of

DIFFERENTIALS 153

curvature so that this is the differential equation of such
lines. If we expand the total differentials we have

du2Spivi>i + 2dudvS(piw2 + Pivv\) + dv2Sp2w2 = 0.

We may also write the equation in the form

dp + xv + ydv = 0 = pidu -\- p2dv + xv + yv\du + yv2dv.

Multiply by (pi + yv\){p2 + yv2) and take the scalar part
of the product, giving

S(pi + yvi)(pi + P2#> = o

= y2Svviv2 + 2ySv{piv2 + ^ip2) + ^2.

The ultimate intersection of the two normals is given by

t = p + dp + yv + y<&>,

that is by yv. Hence we solve for yTv, giving two values
R and Rf which are the principal radii of curvature at the
point. The product and the sum of the roots are re-
spectively

RR' = yy'Tv2 - Tv%- Sw1v2),
R + R' = — 2TvSv(piv2 -\- vip2)/Swiv2.

The reciprocal of the first, and one-half the second divided
by the first, that is,

— Spvivt/v4 and Sv(piv2 + vip2)/Tv*,

are the absolute curvature and the mean curvature of the
surface at the point.

The equation of the lines of curvature may be also written

vSdpvdv = 0 = V-VdpVvdv = VdpV(dv/vv) = VdpdUv.

Hence the direction of dUv is that of a line of curvature,
when du and dv are chosen so that dp follows the line of
curvature. That is, along a line of curvature the change
li

154 VECTOR CALCULUS

in the direction of the unit normal is parallel to the line
of curvature.

When the mean curvature vanishes the surface is a
minimal surface, the kind of surface that a soapfilm will
take when it extends from one curve to another and the
pressures on the two sides are equal. The pressure indeed
is the product of the surface tension and twice the mean
curvature, so that if the resultant pressure is zero, the
mean curvature must vanish. If the radii are equal, as in
a sphere, then the resultant pressure will be twice the
surface tension divided by the radius, for each surface of the
film, giving difference of pressure and air pressure = 4
times surface tension/radius. The difference of pressure
is thus for a sphere of 4 cm. radius equal to the surface
tension, that is, 27.45 dynes per cm.

When a surface is developable the absolute curvature is
zero, and conversely. Surfaces are said to have positive
or negative curvature according as the absolute curvature
is positive or negative.

EXERCISES

1. The differential equation of spheres is

Vp(p - a) = 0.

2. The differential equations of cylinders and cones are respectively

Sva = 0, Sv(p - a) = 0.

3. The differential equation of a surface of revolution is

Sapv = 0.

4. Why is the center of spherical curvature of a spherical curve not
of necessity the center of the sphere?

5. Show how to find the vector to an umbilicus (the radii of curvature
are equal at an umbilicus).

6. The differential equation of surfaces generated by lines that are
perpendicular to the fixed line a is

SVav<pVocv = 0,

where <p is a linear function.

DIFFERENTIALS 155

7. The differential equation of surfaces generated by lines that meet
the fixed line V(p — (3) a = 0 is

SVvV{p - P)a<p{VVV(p - 0)a) = 0.

8. The differential equation of surfaces generated by equal and
similarly situated ellipses is

SV(Va&-p)v(VYa0-p) = 0.

9. Show that the catenoid

p = xi + a cosh x/a(cos 8j + sin 6k)

is a minimal surface, and that the two radii are db Tv, the normal which
is drawn from the point to the axis.

2. Differentiation as to a Vector
1. Definition. Let q = /(p) be a function of p, either
scalar, vector, or quaternion. Let p be changed to p + dt • a
where a is a unit vector, then the change in q is given by

dq= q' - q = f{p+ dt-a) - /(p),
and

dq/dt = Lim [/(p + dta) - f(p)]/dt

as dt decreases. If we consider only the terms in first
order of the infinitesimal scalar dt we can write

dq = dtf(p, a)

in which a will enter only linearly.

In a linear function of a however we can introduce the
multiplier into every term in a and write dta = dp, so that
we have dq a linear function of dp,

dq = f'(p, dp).

It needs to be noted that the vector a is a function of the variable dt,
although a unit vector. The differential of q is of course a function of
the direction of dp in general, but the direction may be arbitrary, or be
a function of the variable vector p. It may very well happen that the
limit obtained above may be different for a given function / according
to the direction of the vector a. In general, we intend to consider the

156 VECTOR CALCULUS

vector dp as having a purely arbitrary direction unless the contrary is
stated.

EXAMPLES

(1) Let

q = " P2.
Then

dq = - [p2 + 2dtS-pa - p2] = - 2dtSpa = - 2Spdp.

Also since q = T2p we have

dq = 2TpdTp=- 2Spdp,
whence

dTp/Tp = Sdp/p, or dTp = - SUpdp.

(2) From the definition we have

d(qr) = dq-r + g-dr,
hence

d(Tp-Up) = dTp-Up+ Tp-dUp = dp

and utilizing the result of the preceding example, we have

dUp/Up = Vdplp.

Also we may write dUp = — Vdpp-p/T3p = pVdpp/T3p
= p~lVpdpjTp, etc. This equation asserts that the dif-
' ferential of Up is the part of the arbitrary differential of p
perpendicular to Up, divided by the length of p, that is,
it is the differential angle of the two directions of p laid off
in the direction perpendicular to p in the plane of p and
dp. In case dp is along the direction of p itself,

dUp = 0.

(3) We have since

d{pp~lp) = dp = dp-p~lp + pd{p~l)p + pp~ldp

= 2dp + pd(p'l)p9

DIFFERENTIALS 157

and thence

dp = — pd{p~l)p,

i.p-i = - p-Hpp-1 = [p-'Spdp - p-WpdpWFp

= p-'dp-p/rp.

That is, the differential of p~l is the image of dp in p divided
by the square of Tp.
Hence

diVap)-1 = (Vap)-lVadp-VapjTWap.

This vanishes if dp is parallel to a.

(4) If x = — a2/p then dir = — a2p~ldppj T2p, and for two
different values of dp, as dip, dip, we have

diir/diTT = p~ldipld\p-p.

Therefore in the process of "inverting" or taking the
"electrical image " we find that the biradial of two dif-
ferential vectors is merely reflected in p. Interpret this.

(5) T- = c is a family of spheres with a and — a as

p — a

limit points. For a differential dp confined to the surface
of any sphere we have then

Sdp[(p + a)-1 - (p - a)-1] = 0.

A plane section through a can be written Syap = 0, in
which Syadp = 0 gives a differential confined to the plane.
Therefore a differential tangent to the line of intersection
of any plane and any sphere will satisfy the equation

Vdp[VVya«p + a)-1 - (p - a)"1)] - 0.

But the expression in the () is a tangent line to any sphere
which passes through A and — A. For the equation of
such a sphere would be

p2 - 2Sadp - a2 = 0

158 VECTOR CALCULUS

where 5 is any vector, hence for any dp along the sphere,
S(p — VaS)dp = 0. But (p + a)-1 — (p — a)-1 is parallel
to a(p2 + «2) - 2pSap and 5(p - Va8)[a(p2 + a2)
— 2piSap] = — Sap[p2 — a2 — 2£pa5]. For points on the
sphere the [] vanishes, hence the vector in question is a
tangent line. Also Vttt is perpendicular to it or r, therefore
the differential equation above shows that the tangent dp
of the intersection of the plane and the sphere of the
system is perpendicular to a sphere through A and —A.
Hence all spheres of the set cut orthogonally any sphere
through A and —A.

(6) The equation SU = e is a familv of tores pro-
p—a

duced by the rotation of a system of circles about their
radical axis. From this we have

SU(p + a)(p-a) = -e,

VU(p + a)(p - a) - V (1 - e2)UVap = a.

Differentiating the scalar equation we have

L P+ OL

+ TJ(p + a)V-^- >U(p - a)l = 0

P — OL J

or

Sadp[(p + a)-1 - (p - a)-1] = 0.

Now in a meridian section a is constant so that

Vdp[(p + a)"1 - (p - a)"1] - 0

and dp is for such section tangent to a sphere through
A and —A.

EXERCISES
1. The potential due to a mass m at the distance Tp is m/Tp in

DIFFERENTIALS 159

gravitation units. Find the differential of the potential in any direc-
tion, and determine in what directions it is zero.

2. The magnetic force at the point P due to an infinite straight
wire carrying a current a is H = — 2h/Vap. Find the differential of
this and determine in what direction, if any, it is zero. For Vdpa = 0,
dH = 0; for dp = dsVa^Vap, dH = - Hds/TV<rP; for dp = dsUVap,
dH - V<rUd8./TV<rp.

3. The potential of a small magnet a at the origin on a particle of
free magnetism at p is u = Sap/T3p. Find the variation in directions

Up, UVap, UaVap.

4. The attraction of gravitation at a point P per unit mass in gravita-
tion units is

a = - Up/T*p.

Find the differential of <x in the directions Up and F/3p.

da = - {pHp - SpSpdp)/T5P; parallel to p, - 2/p3;

perpendicular, UV@p/Tsp.

5. The force exerted upon a particle of magnetism at p by an element
of current a at the origin is

H = - V<xPITsp.

Then dH = {pWadp - 3VaPSpdp)/T5P; in the direction of p, 37a/p3;
in the direction Vap, — VaUVap/T3p.

6. The vector force exerted by an infinitesimal plane current at
the origin perpendicular to a, upon a magnetic particle or pole at p is

a = (ap2 - SpSap)/T*P.

Find its variation in various directions.

2. Differential of Quaternion. We may define differen-
tials of functions of quaternions in the same manner as
functions of vectors. Thus we have T2q — qKq so that

2TqdTq = d(qKq) = [(q + dtUq)(Kq + dtUKq) - qKq]
= dtlqUKq + UqKq]
= qKdq -f- dqKq
= ZSqKdq = 2SdqKq.
That is,

dTq = SdqUKq = SdqUq'1 = TqSdq/q
or

dTq/Tq = Sdq/q.

160 VECTOR CALCULUS

In the same manner we prove the other following formulae.

dUq/Uq = Vdq/q, dSq = Sdq, dVq = Vdq,

dKq = Kdq, S(dUq)/Uq = 0,

dSUq = SUqV(dq/q) = - S(dq/qUVq')TVUq

= TVUqdzq,
dVUq = VUKqV(dqlq),

dTVUq = - SdUqUVq = SUqdzq,

d-q2= 2Sqdq + 2Sq- Vdq + 2Sdq- Vq,

d-q~l = — q~ldqq~l,

d-qaq-1 = - 2V -qdq^qVaq-1 = 2V-dq(Va)q-\

that is, if r = gag-1, then

dr = 2V(dqjq>r) = - 2V(q-dq-l-r)

= 2V(Vdq/q)r - 2q-V 'V{q-ldq-a)q~l
dUVq= V'Vdq/Vq-UVq,
dzq= S[dqKUVq-q)].

We define when 7a = 1

ax = cos • irx/2 + sin • 7nc/2 • a = catf • %tx;
thus

d-a* = tt/2-o:^1^.

If Ta # 1, then

d-ax = dz[log 7W* + tt/2 -ax+1/Ta\,

3. Extremals. For a stationary value of /(p) in the
vicinity of a point p we have ay(p) = 0. If /(p) is to be
stationary and at the same time the terminal point of p
is to remain on some surface, or in general if p is to be subject

*Tait, Quaternions, 3d ed., p. 97.

DIFFERENTIALS 161

to certain conditioning equations, we must also have, if
there is one equation, q(p) = 0, dq(p) = 0, and if there are
two equations, g(p) = 0 and h(p) = 0, then also dg(p) = 0,
dh(p) = 0. Whether in all these different cases /(p) attains
a maximum of numerical value or a minimum, or otherwise,
we will consider later.

EXERCISES

1. g(p) = (p — a)2 -f- «2 = 0, find stationary values of Tp = f(j>).
Differentiating both expressions,

Sdp(p —a) = 0 = Sdpp,

for all values of dp. Hence we must have dp parallel to V • tp where t is
arbitrary, and hence Srp(p — a) = 0, for all values of r. Therefore
we must have Vp(p — a) = 0, or Yap = 0, or p = ya. Substituting
and solving for y,

y = 1 ± a/Tcc, p = a ± aUa. .

2. g(p) = (p — «)2 + a2 = 0. Find stationary values of &/3p.
Sdp(p - a) = 0 = *S/3ap, whence dpP.WjS, £'T,3(p - a) = 0,

7/3(p - a) = 0.

p - a =y0, y = a/T(3, p = a ± at//3.

3. ^(p) = (p — a)2 -f a2 = 0, &G>) = *S/3p = 0, find stationary values
of Tp.

Sdp(P - a) = 0 = Spdp = Spdp, whence S-p0(p — a) « 0 - £pa/S,
and since £p0 = 0, p = yV-fiVafi.

p = V0VaP(l ± V[a2 - S*a0)/TVal3).

4. #(p) = p2 — SapSpp + a2 = 0. Find stationary values of Tp.

£dPp = o = £dp(p - «S/8p - jSflap),

p = x(aS$p + /8/Sap) = (a£/3p + pSap)/(Sa(3 ± Ta0),

whence

Sap = TaSpU/3,

= SpU(3(Ua ± U0)/(SUaU0 T 1).

Substituting in the first equation, we find SpUp, thence p.
5. Sfip = c, >Sap = c', find stationary values of Tp.

SdPp = Sadp = Spdp = 0, p = xa + y$ and

z£a/3 -f 2//32 = c, xa2 + ySafi = c', whence x and y.

162 VECTOR CALCULUS

6. Find stationary values of Sap when (p — a)2 -f- a2 = 0.

Sctdp = 0 = Sdp(p - a);
hence

p = ya = a ± aJ7a
and

Sap = a2 ± aTa.

7. Find stationary values for Sap when p2 — SppSyp + a2 = 0.

Sadp = 0 = £dP(p - )857P - ySfip),
P =xa+ fiSyp + ySfip, etc.

8. Find stationary values of TV8p when

(p - a)2 + a2 = 0.

9. Find stationary values of SaUp when

(p - a)2 + a2 = 0.

10. Find stationary values of SaUpSpUp when

Syp + c = 0.

4. Nabla. The rate of variation in a given direction of
a function of p is found by taking dp in the given direction.
Since df(p) is linear in dp it may always be written in the
form

where $ is a linear quaternion, vector, or scalar function
of dp. In case / is a scalar function, $ takes the form

— Sdpv,

where v is a function of p, which is usually independent of
dp. In case v is independent of the direction of dp, we
call / a continuous, generally differentiable, function.
Functions may be easily constructed for which v varies
with the direction of dp. If when v is independent of dp we
take differentials in three directions which are not in the
same plane, we have

pS - dipd2pd3p = V'dipd2p-Sd3pp + V '• d2pd3p • Sdipp

+ V - d3pdipSd2pp
= — V 'd1pd2p'd3f '— Vd2pd3p-dif

— V-d3pdip-d2f.

DIFFERENTIALS 163

It is evident that if we divide through by Sdipdipdzp, the
different terms will be differential coefficients. The entire
expression may be looked upon as a differential operation
upon/, which we will designate by V. Thus we have

v= V/ =

_ ( Vdipdip - dz + V- d2pdsp • di + V- d^pdip ■ d2) ,, ,
S • dipdipdzp
We may then write

df(P) = - SdpVfip).

If the three differentials are in three mutually rectangular
directions, say i, j, k, then

V = id/dx + jd/dy + kd/dz.

It is easy to find V/ for any scalar function which is gener-
ally differentiate from the equation for df(p) above, that
is, df(p) = — SdpVf. For instance,

VSap = - a, Vp2 = - 2p, VTp = Up,
V(Tp)n = nTpn-lUp = nTpn~2-p, V TVap = TJVap-a,
VSaUp = - p-WUpa, V • SapSpp = - pSd$ - Vap(3,

V-log TVap = -^~,
vap

VT(p - a)-' = - U(p - a)lT\p- a),
VSaUpS(3Up = p-WpVap^P,
Vlog Tp= UP/Tp= -p~\

1

V(ZpA*) = - p~1UVpa =

pUVap

5. Gradient. If we consider the level surfaces of /(p),
/(p) = C, then we have generally for dp on such surface or
tangent to it S dp p = 0 = df(p) where p. is the normal of the

164 VECTOR CALCULUS

surface. Since Sdp\7f — 0 and since the two expressions
hold for all values of dp in a plane

M = *V/,

or since the tensor of p. is arbitrary, we may say V/(p) is
the normal to the level surface of /(p) at p. It is called
the gradient of /(p), and by many authors, particularly in
books on electricity and magnetism, is written grad. p.

The gradient is sometimes defined to be only the tensor
of V/, and sometimes is taken as — V/. Care must be
exercised to ascertain the usage of each author.

Since the rate of change of /(p) in the direction a is
— &*V/(p), it follows that the rate is a maximum for the
direction that coincides with UVf, hence the gradient

V/(p)

gives the maximum rate of change off(p) in direction and size.
That is, TVf is the maximum rate of change of /(p) and
UVfis the direction in which the point P must be moved in
order that /(p) shall have its maximum rate of change.

6. Nabla Products. The operator V is sometimes called
the Hamiltonian and it may be applied to vectors as well
as to scalars, yielding very important expressions. These
we shall have occasion to study at length farther on. It
will be sufficient here to notice the effect of applying V and
its combinations to various expressions. It is to be ob-
served that VQ may be found from dq, by writing dq
= $-dp, then VQ = i$i + j$j + k$k.

For examples we have

Vp = {Vdipdzp-dzp + Vd2pdsp-dip

+ Vdzpdip • d2p) I '(— Sdipd2pd3p)

= - 3

since the vector part of the expression vanishes.

DIFFERENTIALS 165

Vp_1 = — (Vdipd2p-p~1d3pp~1 + •••)/(— Sd1pd2pd3p)
Since

- - P"2.

dUp = V^ • Up, dTp = - SUpdp.

Hence

VUp = 2iV--Up= -~, VTp= Up.
p Tp

Expressions of the form 2F(i, i, Q) are often written
F($ > r> Q)> a notation due to McAulay.

Vap = a,

VfaSfap + cx2S/32p + mSfop) = - 0m + /52a2 + 1830:3),

VFap = 2a, VVap(3 = &xft

VSapVfip = - Sapp + 3/3£a<p - pSa(3,

VVaUp= (a + p^Sap)/ Tp, V • TTap = C/Fap • a,

VTVpVap - (Fap + ap)UVpVap,

VVap/T3p m (ap2 - SpSap)/T5p,

V'UV«P=Tkp> VUVpVap = ^P-,

V(Vap)-i=0, V(g)=0.

EXERCISE

Show that (Fa/3 -<l>y -+- y0y<£>a + Vy<x'3?P)/Sa0y is independent
of a, /3, 7, where $ is any rational linear function (scalar, vector, or
quaternion) of the vector following it. If <*> = S8( ) + 2ai<S/3i( ) the ex-
pression is 5 + S/Siai.

Notation for Derivatives of Vectors
Directional derivative
- SaV, Tait, Joly.
a- V, Gibbs, Wilson, Jaumann, Jung.

Tp -a, Burali-Forti, Marcolongo.

166 VECTOR CALCULUS

Circuital derivative
VaV, Tait, Joly.
a X V, Gibbs, Wilson, Jaumann, Jung.

Projection of directional derivative on the direction.
S-<rlvSau, Tait, Joly.

— > Fischer.
da

Projection of directional derivative perpendicular to the

direction

V-trhi'SV'a, Tait, Joly.

—— * Fischer.
da

Gradient of a scalar

V, Tait, Joly, Gibbs, Wilson, Jaumann, Jung, Carvallo,

Bucherer.
grad, Lorentz, Gans, Abraham, Burali-Forti, Marcolongo,

Peano, Jaumann, Jung. .

— grad, Jahnke, Fehr.

[Fischer's multiplication follows Gibbs, d/dr
d p. , being after the operand, the whole being

dr read from right to left; e.g., Fischer's

Vfl is equiv. to — vSV.]

Gradient of a vector

V, Tait, Joly, Gibbs, Wilson, Jaumann, Jung, Carvallo.
grad, Jaumann, Jung.

-=- > Fischer.
dr

7. Directional Derivative. One of the most important
operators in which V occurs is— SaV, which gives, the

DIFFERENTIALS 167

rate of variation of a function in the direction of the unit
vector a. The operation is called directional differentiating.

SaV'Sfo = - SaP, SaV-p2 = - 2Sap,
SaVTp - SaUp, SaVTp-1 = - Sap/Tp* = UY^p-2,

SaVTVap= 0, SaV-Up= - ^~ •

An iteration of this operator upon Tp~l gives the series of
rational spherical and solid harmonics as follows :

- SaVTp-1 = - Sap/Tp* = UYiTff*,
Sl3VSaVTp-1= (3SapS(3p+ Tp2Sa(3)Tp-5 = 2\Y2Tp~\
SyVSWSaVTp-1 = - (3.5SapS(3pSyp

+ 32S(3ySapTp2)Tp-7 = 3\Y3Tp~\

For an n axial harmonic we apply n operators, giving

Yn = S.(- l)8(2n - 2s)!/[2n-*nl(n - s)l\ESn-28aUpSsa1a2,

0 ^ s^ n/2.

The summation runs over n — 2s factors of the type
SaiUpSoi2Up • - • and s factors of the type SajCtjSotnar - - ,
each subscript occurring but once in a given term. The
expressions Y are the surface harmonics, and the expressions
arising from the differentiation are the solid harmonics
of negative order. When multiplied by Tp2n+1 we have
corresponding solid harmonics of positive order.

The use of harmonics will be considered later.

8. Circuital Derivative. Another important operator is
Va\7 called the circuital derivative. It gives the areal
density of the circulation, that is to say, if we integrate
the function combined with dp in any linear way, around
an infinitesimal loop, the limit of the ratio of this to the
area of the loop is the circuital derivative, a being the normal
to the area. We give a few of its formulae. We may also

168 VECTOR CALCULUS

find it from the differential, for if dQ = $dp, Fa V • Q

VaV • Tp - VaUp, FaV • Tpn = nTpn~2Vapt

VaV - Up = (Sap2 - pSap)/Tp\ VaV-SQp = F/3a,

Fa V • V(3p = a(3+ S-aP, FaV -ft> = 2Sa(3,

FaV • 7Tft> - - V-apUVpp, FaV -p - - 2a,

Fa V • (aiSftp + a2»S/32p + a3S/33p) - Sa(« A + "A

+ a3fo) + FaiFa/3i + Fa2Fa/32 + Fa3Fa/33.

9. Solutions of VQ = 0, V2Q =0. In a preceding
formula we saw that V(Vap)~l = 0. We can easily find
a number of such vectors, for if we apply Sa V to any vector
of this kind we shall arrive at a new vector of the same
kind. The two operators V and Sa V • are commutative
in their operation. For instance we have

d(Vap)~l = - (VapyWadp-iVap)-1;
hence

T = ^V-(Fap)-1 = {Vap)-lV$a>{Vap)-1

is a new vector which gives Vr = 0. The series can easily
be extended indefinitely. Another series is the one de-
rived from Up/T2p. This vector is equal to p/T3p, and its
differential is

(-p2dp+SSdpp.p)/T%

The new vector for which the gradient vanishes is then

(-ap2+3Sap-p)/7V

The latter case however is easily seen to arise from the
vector V Tp~l, and hence is the first step in the process of
using V twice, and it is evident that S72Tp~l = 0. So also
the first case above is the first step in applying V2 to log
TVap-a~l so that V2(log TVap-a) = 0. Functions of p
that satisfy this partial differential equation are called

DIFFERENTIALS 169

harmonic functions. That is,/(p) is harmonic if V2/(p) = 0.
Indeed if we start with any harmonic scalar function of p
and apply V we shall have a vector whose gradient van-
ishes, and it will be the beginning of a series of such vectors
produced by applying &*iV, Sa2 V, • • • to it. However we
may also apply the same operators to the original harmonic
function deriving a series of harmonics. From these can
be produced a series of vectors of the type in question.
V2 • F(p) is called the concentration of F(p) . The concentra-
tion vanishes for a harmonic function.

EXERCISES

Show that the following are harmonic functions of p:
1. Tp-1 tan"1 Sap/Spp,

where a and /? are perpendicular unit vectors,

2.

Tf* log tan ^ Z £

3.
where

and

tan-1 Sap/S/3p

Sa(3 = 0
a2 = £2 = _ 1#

4.

logtan^ z - •

£j CL

10. Harmonics. We may note that if u, v are two scalar
functions of p, then

V -uv = u Vfl + v\7u
and thus

V2-uv = u\7h + vV2u + 2SVuVv.

Hence the product of two harmonics is not necessarily
harmonic, unless the gradient of each is perpendicular to
the gradient of the other.
Also if u is harmonic, then

\72-uv = u\72v + 2SVu\7v.

12

170 VECTOR CALCULUS

If u is harmonic and of degree n homogeneously in p, then
w/7p2n+I is a harmonic of degree — (n + 1). For

V2(fp2n+1)-1 . V[_ (2n+ l)prp-2n-3]

= - (2n+ l)(2n)Tp~2n-3
and
SVuVTp-2"-1 = - (2n+ l)Tp-2n-*SVup

= (2n+ l)(2n)uTp-2n-*;
hence

V2-u/Tp2n+1 - 0.

In this case w is a solid harmonic of degree n and uTp~2n~l
is a solid harmonic of degree — n — 1. Also uTp"11 is a
corresponding surface harmonic. The converse is true.
EXAMPLES OF HARMONICS

Degree n = 0; <p = tan-1 — -

£>pp

where Sc& = 0, a2 = /32 = - 1;

^ = log cot ^/ -a2 = - 1;
a

S-a(3UpSapS(3p/V2-a(3p;

Sa(3UpS(a + 0)pS(a - /3)p/F2a/3p.

The gradients of these as well as the result of any opera-
tion Sy V are solid harmonics of degree — 1, hence multiply-
ing the results by Tp[n = 1, 2n — 1 = 1] gives harmonics
again of degree 0. These will be, of course, rational
harmonics but not integral.

Taking the gradient again or operating by $71 V any
number of times will give harmonics of higher negative
degree. Multiplying any one of degree — n by Tp2n~1
will give a solid harmonic of degree n — 1.

Degree n = — 1. Any harmonic of degree 0 divided by
Tp, for example,

1/Tp, ip/Tp, f/Tp, Saf3UpSaUpS(3p/V2a(3p, • • • ,

DIFFERENTIALS 171

Degree n = — 2.

SaUp/p2, <pSa(3Up/p2, xPSa(3Up/p2 + P"2 • • • .
Degree n = 1.

Sop, *>&*ft>, ^Softa + 7p • • • .

Other degrees may easily be found.

11. Rational Integral Harmonics. The most interesting
harmonics from the point of view of application are the
rational integral harmonics. For a given degree n there
are 2n + 1 independent rational integral harmonics. If
these are divided by Tpn we have the spherical harmonics
of order n. When these are set equal to a constant the
level surfaces will be cones and the intersections of these
with a unit sphere give the lines of level of the spherical
harmonics of the given order. A list of these follow for
certain orders. Drawings are found in Maxwell's Electricity
and Magnetism.

Rational integral harmonics, Degree 1. Sap, S(3p, Syp,
a, ft, y a trirectangular unit system.

Degree 2. SapS(3p, SfoSyp, SypSap, 3S2ap + p2, S2ap

- s2pP.

These correspond to the operators 7p5[£27V, SyVSaV,
SyVSPV, S(a + 0) VS(a - 0) V, SaVSQV] on Tp'K

Degree 3. Representing Sap by — x, Sfip by — y, Syp by
— z, SaV by — Dx, S(3V by — Dy, SyV by — Dz we have

2z3 — 3x2z — 3y2z, 4:Z2x — x* — y2x, A.z2y — x2y — y3,

x2z — y2z, xyz, xz — 3xy2, 3x2y — y3

corresponding to

7)3 7)3 7)3 7) 3 _ 7) 3 7) 3 _ Q7) 3

^ zzz ) -lszzx , Lf zzy , ^xxz > J^xyz , ^xxx > OUXyy ,

7) 3 _ Q7) 3

■LSyyy j OJ^xxy •

172 VECTOR CALCULUS

Degree 4.

3z4 + 3y4 + 8z4 + 6*y - 24z2z2 - 24yV,

*z(4z2 - Sx2 - 3y2), yz(4z2 - 3^ - 3i/2)

(^ _ y2)(6z2 — x2 — y2), xy(6z2 — x2 — y2),

xz(x> - Sy2), yzQx2 - y2), x* + y* - My2,

xyix2 - y2)

7) 4 7) 4 7) 4
is zzzz ) *-* zzzx y -L/zzzy

D 4 -

■LS ZZXX

. T) 4 7) 4 7) 4 _ OT) 4

*s zzyy j M* zzxy i J^xxxz *->±sXyyz j

7) 4

1Jyyyz

_ Q7) 4 7) 417) 4_ ft/) 4
oiyxxyz ) Uxxxx T ^ yyyy ^^xxyy y

D 4 — 7) 4

J^xxxy ^xyyy •

The curves of the intersections of these cones with the
unit sphere are inside of zero-lines as follows :

Degree 1. Equator, standard meridian, longitude 90°.

Degree 2. Latitudes ± sin-1 JV 3, equator and standard
meridian, equator and longitude 90°, longitude ± 45°.
Standard meridian and 90°.

Degree 3. Latitudes 0°, db sin-1 V 0.6, latitudes ± sin-1
V 0.2 and standard meridian, latitudes ± sin-1 V 0.2 and 90°
longitude, equator, longitude ± 45°, equator, longitudes
0°, 90°. Longitudes ± 30°, 90°, longitudes ± 60°, 0°.

12. Variable System of Trirectangular Unit Vectors.
We will consider next a field which contains at every point
a system of three lines which are mutually perpendicular.
That is, the lines in one direction are given by a, say, at the
same point another set by ft and the third set by y. Each
is a given function of p, subject to the conditions

a/3 = 7, /?7 = a, ya = /?, a2 = (32 = y2 = — 1.

For example, in the ordinary congruence, a being the unit
tangent at any point of one line of the congruence, then
the normal and the binormal are determined and would
be ]S and 7. However /? and 7 may be other perpendicular

DIFFERENTIALS 173

lines in the plane normal to a. If we follow the vector
line for /3 after we leave the point we shall get a determinate
curve, provided we consider a to be its normal. We may
however draw any surface through the point which has
a for its normal and then on the surface draw any curve
through the point. All such curves can serve as ft curves
but a might not be their principal normal. It can happen
therefore that the j8 curves and the y curves may start out
from the point on different surfaces. However a, (3, and y
are definite functions of the position of the point P, with
the condition that they are unit vectors and mutually
perpendicular.

If we go to a new position infinitesimally close, a becomes
a + da, ft becomes fi + dp, and y becomes y + dy. The
new vectors are unit vectors and mutually perpendicular,
hence we have at once

S-ada = S-pdp = S>ydy = 0, Sadp = - S(3da, n .
Spdy = - Sydp, Syda = - Sady. {L)

These equations are used frequently in making reductions.
We have likewise since a2 = — 1,

Va-a = - VW, V/3-/3 = - VW, (2)

vy-t = - v'rr'j

where the accent on the V indicates that it operates only
on the accented symbols following. Similarly we have

Va-j8 + V(3-a= - V'a0' - V'j&x', etc. (3)

We notice also that

S-a(SQV)a = 0,
S-a(SQV)0 = - S-p(S()V)a, etc. (4)

We now operate on the equation y = afi with V, and

174 VECTOR CALCULUS

remember that for any two vectors X/x we have X/x = — juX
+ 2<SX/x, whence

V7 = Va-j3 + V'aP' = Va-/3 - V/3-a + 2V'Sa(3'. (5)

The corresponding equations for the other two vectors are
found by changing the letters cyclically.
Multiply every term into y and we have

Vt-7 = Vo-a + Vj8-|8 + 2V'Sct(3'-y. (G)

If now we take the scalar of both sides we have

SyVy = SaVa + Sj8V0 + 2SyV'Sa(3'. (7)

We set now

2p = + &*Va + SjSVjS + #7 Vt (8)

and the equation (7) gives, with the similar equations
deduced by cyclic interchange of the letters,

SyVSctP - - SyV'Sa'Q = - p + S7V7,
SaV'SPy' m - SaV'Sfi'y = - P + 5a V«,
SpVSya' = - SpV'Sy'a = - p + S0V/3,
- S-Tf- 5a V • y] = 5a V • £77' = |&* V ■ 72 = 0, ( j
- 5-a[- 5aV-7] = - SaV-Sa'y

= Sy(— SaV -a) = Sy(u(3 + vy) = — v.

That is to say, the rate of change of y, if the point is moved
along a, is ]8(5aVa — p). Likewise

dfi/ds = — 7(— p + 5aVa)— -ya.

The trihedral therefore rotates about a with the rate
(p — SaVa) as its vertex moves along a. Now we let

ta = + p - SaVa. (10)

We may also write at once, similarly,

h - + V - S0VA *7 = + p - 57V7, (10)
from which we derive

t + V+ <»-+■* (ID

DIFFERENTIALS 175

It is also evident that

*. + U = SrVy, tfi + *, - SaVa, /7 + /a = 5/3 V/3. (12)

The expressions on the left hold good for any two per-
pendicular unit vectors in the plane normal to the vector
on the right, and hence if we divide each by 2 and call the
result the mean rotatory deviation for the trajectories of the
vector on the right, we have

TjSctVct = mean rotatory deviation for a.

Again the negative rotation for the 0 trajectory gives
what we have called previously the rotatory deviation of a
along j3. Hence, as a similar statement holds for y, the
mean rotatory deviation is one half the sum of the rotatory
deviations. Hence %Sa\7a is the negative rate of rotation
of the section of a tube of infinitesimal size, whose central
trajectory is a, about a, as the point moves along a. Or
we may go back to (9) and see that

SaVa = (+ p ~ SPVB) + (+ V ~ SyVy)

= - SpV'Sya' + SyV'Sfa',

which gives the rotatory deviations directly.
The scalar of (5) and the like equations are

SVa = SyVP - Sj3\7y, SVP = SaVy - SyVa,

(13)

SVy = SfiVa - SaVP,
We multiply next (5) by a and take the scalar, giving

SyVa = - SaV'Sfia' = SaV'Sa(3f,
SfiVa m - SaVSay* = SaV'Sya',
SaVP = - SpV'Sy? = St3V'S(3y',
£TV/3 = - SpV'Spa' = S(3V'Sa(3',
SfiVy = - SyV'Say' = SyV'Sya',
SaVy = - SyV'Sy(3' m SyV'Sfiy'.

(14)

176 VECTOR CALCULUS

We can therefore write

SVa = - SWSPa' - SyV'Sya',

that is SVa equals the negative sum of the projection of
the rate of change of a along (3 on /3, and the rate of change
of a along y on y. But these are the divergent deviations
of a and hence — SVa is the geometric divergence of the
section. It gives the rate of the expansion of the area of
the cross-section of the tube around a. We may write the
corresponding equations of /8 and y.
Again we have

FVa = — aSaVa — (3S(3Va — ySyVoc

= cx(ta - v) - PSy(SaV-a) + ySp(SaV-a)
= a(ta — p) — Va(SaV-a).

Now from the Frenet formulae

— Sa'V 'Ol = cav,

where ca is the curvature of the trajectory and v is the
principal normal. Hence

Wa = a(ta - p) + CJh (15)

where /i is the binormal of the trajectory. We find there-
fore that VVd consists of the sum of two vectors of which
one is twice the rate of rotation of the section or an elemen-
tary cube about a, measured along a, and the other is twice
the rate of rotation of the elementary cube about the
binormal measured along the binormal.* But we will see

* This should not be confused with the rotation of a rigid area mov-
ing along a curve. The infinitesimal area changes its shape since each
point of it has the same velocity. As a deformable area it rotates (i.e.
the invariant line of the deformation) with half the curvature as its
rate. The student should picture a circle as becoming an ellipse,
which ellipse also rotates about its center.

DIFFERENTIALS 177

later that this sum is the vector which represents twice the
rate of rotation of the cube and the axis as it moves along
the trajectory of a. Hence this is what we have called
the geometric curl.

We may now consider any vector a defining a vector
field not usually a unit vector. Since a = TaUa, we have

SVa = SUaVTa + TaSvUa.

The last term is the geometric convergence multiplied by
the length of a, that is, it is the convergence of a section
at the end of a. The first term is the negative rate of
change of TV along a. The two together give therefore
the rate of decrease of an infinitesimal volume cut off from
the vector tube, as it moves along the tube. In the lan-
guage of physics, this is the convergence of a. Similarly
we have

Wa= VvTaU<r+ TaWUa.

The last term is the double rate of rotation of an elementary
cube at the end of a, while the first term is a rotation about
that part of the gradient of Ta which is perpendicular to
Ua. It is, indeed, for a small elementary cube a shear of
one of the faces perpendicular to Ua, which gives, as we
have seen, twice the rate of rotation corresponding. Con-
sequently VVa is twice the vector rotation of the elemen-
tary cube.

EXAMPLES

(1) Show that

aSVa + (3S V0 + yS Vt

= - VaWa - V(3WP - VyV\7y.

(2) Show that if dipt) is zero VaWa = 0. This is the
condition that the lines of the congruence be straight. It
is necessary and sufficient.

178 VECTOR CALCULUS

(3) Let Wot - f, - SaVa « z, then Tf = V [c2 + *%

fi = — &x V • £ = #ia + c^/3 + c%yt where the subscript
1 means differentiation as to s, that is, along a line of the
congruence.

- S^ - cip; a! = cr'Sfei + x,
or

This gives the torsion in terms of the curl of a and its
derivative.

(4) If the curves of the congruence are normals to a set
of surfaces, then

a = UVu and V« = V2u/TVu - V(l/TVu)-Vu.

Hence we have at once SaVa = 0 = x. This condition
is necessary and sufficient.

(5) If also VaWoi — 0, we have a Kummer normal
system of straight rays. In this case by adding the two
conditions, aV\/a = 0, that is, Wot = 0. This condition
is also necessary and sufficient.

(6) If the curves are plane, «i = 0 or Sa\7a = $/3V/3
+ SyVy or $/?£i = — xci, which is necessary and suffi-
cient.

(7) If further they are normal to a set of surfaces S8VP
+ SyVy = 0 = jS|8f i. The converse holds.

(8) If Ci is constant, Sy£i = 0 and conversely.

If also plane, and therefore circles, #/3£i = 0 or £i = X\a
+ C\x(3. This is necessary and sufficient.
For a normal system of circles we have also

VVa = const = C\y.

(9) For twisted curves of constant curvature £i = — ciaifi.

differentials 179

Notations
Vortex of a vector
VVu, Tait, Joly, Heaviside, Foppl, Ferraris.

V X u, Gibbs, Wilson, Jaumann, Jung.

curl u, Maxwell, Jahnke, Fehr, Gibbs, Wilson, Heaviside,

Foppl, Ferraris. Quirl also appears.
[Vm], Bucherer.

rot u, Jaumann, Jung, Lorentz, Abraham, Gans, Bucherer.
J rot u, Burali-Forti, Marcolongo.

— ; — , Fischer.
dr

Vort u, Voigt.

(Notations corresponding to VVu are also in use by

some that use curl or rot.)

Divergence of a vector

— SVu, Tait, Joly. S\7u is the "convergence" of Max-

well.

V • u, Gibbs, Wilson, Jaumann, Jung.

div u, Jahnke, Fehr, Gibbs, Wilson, Jaumann, Jung,
Lorentz, Bucherer, Gans, Abraham, Heaviside, Foppl,
Ferraris, Burali-Forti, Marcolongo.

\7u, Lorentz, Abraham, Gans, Bucherer.

— ~ , Fischer.
dr

Derivative dyad of a vector

- SQV-u, Tait, Joly.

• Vw, Gibbs, Wilson.

• V ; u, Jaumann, Jung.

du

-p=t Burali-Forti, Marcolongo.

aJr

— , Fischer.
dr

Du - , Shaw.

180 VECTOR CALCULUS

Conjugate derivative dyad of a vector

— VS«(), Tait, Joly.
Vm-, Gibbs, Wilson.
V; u-f Jaumann, Jung.

Ki(), Burali-Forti, Marcolongo.

-j-, Fischer.
drc

Du-,Shaw.

Planar derivative dyad of a vector
WVuQ, Tait, Joly.
VX(mX 0), Gibbs, Wilson.
V *u, Jaumann, Jung.

du
CK , Burali-Forti, Marcolongo.

— x(Du), Shaw.

Dispersion. Concentration

— V2, Tait, Joly. V2 is the "concentration" of Maxwell.
V2, Lorentz, Abraham, Gans, Bucherer.

V-V, Gibbs, Wilson, Jaumann, Jung.
div grad, Fehr, Burali-Forti, Marcolongo.

— div grad, Jahnke.

A2, for scalar operands, 1^, ,*,, A. %- .
A/, for vector operands, jBurah-Forti, Marcolongo.

-7-5 > Fischer.
dr

Dyad of the gradient. Gradient of the divergence

— VSV, Tait, Joly.
VV-, Gibbs, Wilson.
V; V, Jaumann, Jung.

grad div, Buroli-Forti, Marcolongo.

DIFFERENTIALS 181

Planar dyad of the gradient. Vortex of the vortex

VVVV(), Tait, Joly.

V*V, Jaumann, Jung.

rot2, Lorentz, Bucheoer, Gons, Abraham.

curl2, Heaviside, Foppl, Ferraris.

rot rot, Burali-Forti, Marcolongo.

13. Vector Potential, Solenoidal Field. If £ = VVv,
then we say that a is a vector potential of £. Obviously

£v£ = SV2<r = 0.

The vector potential is not unique, since to it may be added
any vector of vanishing curl. When the convergence of a
vector vanishes for all values of the vector in a given region
we call the vector solenoidal. If the curl vanishes then
the vector is lamellar.

We have an example of lamellar fields in the vector field
which is determined by the gradient of any scalar function,
for WVu = 0.

In case the field of a unit vector is solenoidal we see from
the considerations of § 12 that the first and second divergent
deviations of any one of its vector lines are opposite. If
then we draw a small circuit in the normal plane of the
vector line at P and at the end of dp a second circuit in
the normal plane at p + dp, and if we project this second
circuit back upon the first normal plane, then the second
will overlie the first in such a way that if from P a radius
vector sweeps out this circuit then for every position in
which the radius vector must be extended to reach the
second circuit there is a corresponding position at right
angles to it in which it must be shortened by an equal
amount. It follows that the limit of the ratio of the areas
of the two circuits is unity. Hence if such a vector tube is
followed throughout the field it will have a constant cross-

182 VECTOR CALCULUS

section. In the general case it is also clear that SVcr gives
the contraction of the area of the tube.

When <r is not a unit vector then we see likewise that
SVcr by § 12 has a value which is the product of the con-
traction in area by the TV -f- the contraction of TV multi-
plied by the area of the initial circuit. Hence SVv repre-
sents the volume contraction of the tube of a for length TV
per unit area of cross-section. When the field is solenoidal
it follows that if TV is decreasing the tubes are widening
and conversely.

For instance, S\/Up = — 2/Tp signifies that per unit
length along p the area of a circuit which is normal to p
is increasing in the ratio 2/Tp, that is, the flux of Up is
increasing at the rate of 2/ Tp along p. Also £ • Vp = — 3
indicates that an infinitesimal volume taken out of the
field of p is increasing in the ratio 3. Of this the increase 2
is due to the widening of the tubes, as just stated, the
increase 1 is due to the rate at which the intensity of the
field is increasing. If the field is a velocity field, the rate
of increase of volume of an infinitesimal mass is 3 times
per second.

It is evident now if we multiply SVo" by a differential
volume dv that we have an expression for the differential
flux into the volume. If a is the velocity of a moving mass
of air, say unit mass, then SV<? is the rate of compression
of this moving mass, and SVcrdv is the compression per
unit time of this mass, and fffSVcrdv is the increase
in mass per unit time of matter at initial density or com-
pression per unit time of a given finite mass which occupies
initially the moving volume furnishing the boundary pf
the integral.

If r is the specific momentum or velocity of unit volume
times the density, then SVr is the condensation rate or

DIFFERENTIALS 183

rate of increase of the density at a given fixed point, and
SVrdv is the increase in mass in dv per unit time. Hence
SffSVrdv is the increase in mass per unit time in a
given fixed space.
Since

1

a — -t
c

where c is density at a point,

SVo- = --SVct + -SVt
cr c

e, „ i . B log c d log c

= _S(TV.logc+_JL_ = _iL

= total relative rate of change of density

due to velocity and to time,
= relative rate of change of density at a
moving point.
SVc-dv= increase in mass of a moving dv divided
by the original density.
fffSVv-dv = increase in mass in a moving volume per
unit of time divided by original density,
= decrease in volume of an original mass.

For an incompressible fluid SVcr = 0 or a is solenoidal,
and for a homogeneous fluid SVt = 0 or t is solenoidal.
In water of differing salinity #Vcr = 0, SVr =\= 0. We
have a case of constant r in a column of air. If we take
a tube of cross-section 1 square meter rising from the ocean
to the cirrus clouds, we may suppose that one ton of air
enters at the bottom, so that one ton leaves at the top, but
the volume at the bottom is 1000 cubic meters and at the
top 3000 cubic meters. Hence the volume outflow at the
top is 2000 cubic meters. In the hydrosphere a and r

184 VECTOR CALCULUS

are solenoidal, in the atmosphere r is solenoidal. We
measure a in m?/sec and r in tons/ra2 sec. At every sta-
tionary boundary <r and r are tangential, and at a surface
of discontinuity of mass, the normal component of the
velocity must be the same on each side of the surface, as
for example, in a mass of moving mercury and water.

It is evident that if a vector is solenoidal, and if we
know by observation or otherwise the total divergent devia-
tions of a vector of length TV, then the sum of these will
furnish us the negative rate of change of TV along a.
Thus, if we can observe the outward deviations of r in the
case of an air column, we can calculate the rate of change of
TV vertically. If we can observe the outward deviations of
a tube of water in the ocean we can calculate the decrease
in forward velocity.

EXERCISES.

1. An infinite cylinder of 20 cm. radius of insulating material of
permittivity 2 [farad/cm.], is uniformly charged with l/207r electrostatic
units per cubic cm. Find the value of the intensity E inside the rod,
and also outside, its convergence, curl, and if there is a potential for
the field, find it.

2. A conductor of radius 20 cm. carries one absolute unit of current
per square centimeter of section. Find the magnetic intensity H inside
and outside the wire and determine its convergence, curl, and potential.

14. Curl. We now turn our attention to another meaning
of the curl of a vector. We can write the general formula
for the curl

W<t= -aSUaVUa- pSyVTa + y(cT(T+ SfiVTa)

Let Ua = a'. These terms we will interpret, one by one.
It was shown that the first term is a multiplied by the sum
of the rotational deviations of <r' . But if we consider a
small rectangle of sides t)dt = dip and rdu = d2p, then the
corresponding actual deviations are

Sdipd2af and — Sd2pdia'

DIFFERENTIALS 185

and the sum becomes

Sdipdtff' — Sd2pdi<r'.

But d2a' is the difference between the values of a' at the
origin and the end of d2p, and to terms of first order is the
difference of the average values of a' along the two sides
dip and d\p + d2p — dip. Likewise dia is the difference
between the average values of a' along the side d2p and its
opposite. Hence if we consider Sdpa' for a path consisting
of the perimeter of the rectangle, the expression above is
the value of this Sdpa' for the entire path, that is, is the
circulation of <j' around the rectangle. Hence the coefficient

- SUaVUa

is the limit of the quotient of the circulation around dip d2p
divided by dtdu or the area of the rectangle.

If we divide any finite area in the normal plane of a into
elementary rectangles, the sum of the circulations of the
elements will be the circulation around the boundary, and
we thus have the integral theorem

fSdpa = ffSdipd2PV\7<j

when Vdipd2p is parallel to Fy<r. The restriction, we shall
see, may be removed as the theorem is always true.
The component of V\7<r along a is then

— Ua Lim j^Sdpcr/area of loop

as the area decreases and the plane of the loop is normal to a.
Consider next the term — (3SyV Ta. It is easy to reduce
to this form the expression

[- S*'(SyV)<r + Sy(S&'V)<r][- j8]. > id ;
But this is the circulation about a small rectangle in the

13

ISC. VECTOR CALCULUS

plane normal to /?. Hence the component of VVcr in the
direction 0 is

— (3 Lim J'Sdpff/aresi of loop in plane normal to /?.

Likewise the other term reduces to a similar form and the
component of V\7<r in the direction 7 is

— 7 Lim tfSdpa/sLYea of loop in plane normal to 7.

It follows if a is any unit vector that the component of
V\7(T along a is

— a Lim JfSdpa/sLfesL of loop in plane normal to a

as the loop decreases. The direction of UVS/a is then
that direction in which the limit in question is a maximum,
and in such case TV\7a is the value of the limit of the cir-
culation divided by the area. That is, TVS/v is the maxi-
mum circulation per square centimeter.

Another interpretation of VV<? is found as follows: Let
us suppose that we have a volume of given form and that a
is a velocity such that each point of the volume has an inde-
pendent velocity given by a. Then the moving volume will
in general change its shape. The point which is originally
at p will be found at the new point p + cr(p)dt. A point
near p, say p + dp, will be found at p + dp + a(p + dp)dt,
and the line originally from p to p + dp has become instead
of dp,

dp -f- dt[a(p + dp) — <r(p)] = dp — SdpV 'vdt.

But this can be written

dp' = dp- [W-^'dpa' + idpSVo- - iV(W(r)dp]dt.

This means, however, that we can find three perpendicular
axes in the volume in question such that the effect of the

DIFFERENTIALS 187

motion is to move the points of the volume parallel to these
directions and to subject them to the effect of the term

dp + iV(W(r)dp dt.

Now if we consider an infinitesimal rotation about the
vector e its effect is given by the form (du being half of the
instantaneous angle)

(1 + edu)p(l — edu) = p + 2Vepdu;

hence the vector joining p and p + dp will become the vector
joining p + 2Vepdu and p + 2Vepdu + dp -f- 2Vedpdu,
that is, dp becomes dp + 2Vedp du. We find therefore
that the form above means a rotation about the vector
UVV<r of amount \TV\7adt, or in other words V\/a,
when a is a velocity, gives in its unit part the instanta-
neous axis of rotation of any infinitesimal volume moving
under this law of velocity, and its tensor is twice the angu-
lar velocity. For this reason the curl of a is often called the
rotation. When V\/<r = 0, a has the form a = \/u, and u
is called a velocity potential. If a is not a velocity, we
still call u a potential for a.

EXERCISES.

1. If a mass of water is rotated about a vertical axis at the rate of
two revolutions per second, find the stationary velocity. What are the
convergence and the curl of the velocity? Is there a velocity potential?

2. If a viscous fluid is flowing over a horizontal plane from a central
axis in such way that the velocity, which is radial, varies as the height
above the plane, study the velocity.

3. Consider a part of the waterspout problem on page 50.

15. Vortices. Since VVc is a vector it has its vector
lines, and if we start at any given point and trace the vector
line of FVo" such line is called a vortex line. The field of
FVc is called a vortex field. If a vector is lamellar the
vector and the field are sometimes called irrotational. The

188 VECTOR CALCULUS

equation of the vortex lines is

VdpWa = 0 - 8dp V a - V'Sdpa' - - da - V'Sdpa'.

The rate of change of a then along one of its vortex lines is
— V'Saa'. Since SvV^a — 0, the curl of a is always
solenoidal, that is, an elementary volume taken along the
vortex lines has no convergence but merely rotates.

The curl of the curl is VvVVa = VV — S/SVa and
thus if a is harmonic the curl of the curl is the negative
gradient of the convergence, and if the vector is solenoidal,
the curl of the curl is the concentration VV.
EXERCISES

1. If Sa<r = 0 = SaV '<r, and if we set <r = V-ar, and determine X
so that V-X" = t, then Xa is a vector potential of the vector <r.

2. Determine the vector lines in the preceding problem for a. Also
show that the derivative of X in any direction perpendicular to a is
equal to the component of a perpendicular to both. What is V2A^?

3. If a = wy and — Sy V • w = 0, then either Xa or F/3 will be
vector potentials of <r where (iy = a and all are unit vectors and
SyV'X =0 = SyVY.

4. If the lines of <r are circles whose planes are perpendicular to y
and centers are on p = ty, and To = f(TVyp), then any vector parallel
to y whose tensor is F(TVyp), where — f = dF/dTVyp is a vector
potential of a. Is a solenoidal?

5. If the lines of <r are straight lines perpendicular to y and radiating
from p = ty and T<r. = f(TVyp), then what is the condition that <r be
solenoidal? If Ta = /(tan-1 TVyp/Syp) a cannot be solenoidal.

6. If a =/(*Sap, S0p)-Vyp-y, then what is FV<r? Show that if/
is a function of tan-1 Sap/Spp, that SypVf is a function of the same
angle, but if / is a function of TVyp, SypV •/ = 0 and no vector of
the form a = f(TVyp)Vyp-y can be a potential of yTVyp. If
M = Sap/Sfip, then/0*) = - ./V0*)eW0*2 + 1).

7. What are the lines of a = f(Sap, Sfip) Vyp and what is the curl?
If / is a function of TVyp, so is the curl, and if

F{TVyp) = (TVyp)-2fTVyp<pTVypdTVyp

then F-TVyp is a vector potential of the solenoidal vector y<pT{Vyp).
If / is a function of p. the curl is a function of p., and \f(ji) Vyp is a vector
potential of 7/O*).

8. If <r is solenoidal and harmonic the curl of its curl is zero. If its

DIFFERENTIALS 189

lines are plane and it has the same tensor at all points in a line per-
pendicular to the plane, then it is perpendicular to its curl.

9. The vector <r = f- Up, where / is any scalar function of p, is not
necessarily irrotational, but SaVv = 0.

10. If a vector is a function of the two scalars S\p, Sup where X, p.
are any two vectors (constant), or if S\p = 0, then what is true of

11. If S<rV<r 4= 0, show that if F is determined from S\7<rVF
= — SaX7 9 then F is the scalar potential of an irrotational vector r
which added to <r gives a vector a', &cr'V V = 0. Is the equation for
F always integrable?

12. The following are vectors whose lines form a congruence of
parallel rays f(p)a, f(Sap)a, f(Vap)a, [where/ is a scalar function], which
are respectively neither solenoidal nor lamellar, lamellar, solenoidal.
The case of both demands that To = constant.

13. Examples of vectors of constant intensity but varying direction
are

o- = aUp, aVocp +«V(62 - a2V2ap).

Determine whether these are solenoidal and lamellar.

14. If the lines of a lamellar vector of constant tensor are parallel
rays, it is solenoidal. If the lines of a solenoidal vector are parallel
straight lines, it is lamellar.

15. An example of vectors whose convergences and curls are equal
at all points, and whose tensors are equal at all points of a surface, are

a(x + 2yz) + &(y + Szx) + xyy, and 2yza + Szx/3 -f- y(xy + 2z)
and the surface is

x2 + y2 - z2 + 6xyz = 0.

Therefore vectors are not fully determined when their convergences and
curls are given. What additional information is necessary to determine
an analytic vector which does not vanish at oo .' Determine a vector
which is everywhere solenoidal and lamellar and whose tensor is 12
for Tp m oo .

16. Show that

— eV2<Z = limr=0 [average value of q over a sphere of radius r, less the

value at the center] divided by r2.

— \V2q = average of (- SaV)2q in all directions a.

— xVV2g = limr=o [excess of average value of q throughout a small

sphere over the value at the center} divided by r2.

17. Show by expansion that

a(p + 8p) = a(p) - S8pX7 -(r(p)

- VSP[- Sa8p + ±S8pVPSa8p] - W8pVSJ p*
= VVSP[~ |F5pa + iSSpVpVSpa] - ±8pSV P<r.

190 VECTOR CALCULUS

i

The first expansion expresses <r in the vicinity of p in terms of a gradient
of a scalar and an infinitesimal rotation. The second expresses a in
the form of a curl and a translation.

18. Show that for any vector <r we have

£V(W'V"&r\r"<r/7V) =0,

where the accents show on what the V acts, and are removed after the
operation of the accented nabla. The unaccented V acts on what is
left. (Picard, Traits, Vol. I, p. 136.)

19. If a, <r2 are two functions of p, and d<n = <pi(dp),da2 = widp),
show that

&<riV -SaiV — S<r2V -SaiV = S(<pi<r2 — <p2<Ti)^7 .

16. Exact Differentials. If the expression Sadp is the
differential of a function u(p), then it is necessary that
Sadp = — SdpVu, for every value of dp, which gives

a = — Vw.

When a is the gradient of a scalar function of u(p), u is
sometimes called a force-function. It is evident at once
that

VS7<r = 0, or £FOV)cr = 0 for every v.

This is obviously a necessary condition that Sadp be an
exact differential, that is, be the differential of the same
expression, u, for every dp. It is also sufficient, for if
VVa = 0, it will, be shown below that a = Vu, and
SVudp = — du.

In general if Q(p) is a linear rational function of p,
scalar or vector or quaternion, then to be exact, Q(dp) must
take the form

Q(dp) — — SdpV -R(p) for every dp.
Hence formally we must have the identity

C()= -S()V-R(p).
But if we fill the ( ) with the vector form VvV , we have
Q(Vi>S7) = 0 for every v.

DIFFERENTIALS 191

This may be written in the form

Q'VV'l ) .- 0 identically.

EXERCISES

1. Vadp is exact only when a = a a constant vector. For VaV\7 v = 0
for every v, that is S\(vSS7p- — VSav) = 0 for every X, v, and for X
perpendicular to v therefore SXS/Sav = 0, or Sdav = 0 for every v
perpendicular to the dp that produces da. Again if X = v,

SV* + SvVSav = 0,

for every v. Therefore S\/a = 0 and Sv\7 Sav = 0, or Sdav = 0 for
every dp in the direction of v. Hence da = 0 for every dp and a = a
a constant.

2. Examine the expressions

S^, V(Vap)dp, F.&.

Integrating Factor
If an expression becomes ezactf &?/ multiplication by a
scalar function of p, let the multiplier be m. Then

mQ(W) = 0,
where V operates on m and Q, or

QWm() + mQVV() = 0,

where V operates on m only in the first term and on Q
only in the second. This gives for Sadp

SaVmi ) + mS( )Vo- = 0, or VaVm + mVV<r = 0.

This condition is equivalent however to the condition

Sa\7<7 = 0.

Conversely, when this condition holds, we must have

VVa- = V(tt,

where r is arbitrary, hence StVv = 0, and Sa\7r = 0.
But r is any variable vector conditioned only by being

192 VECTOR CALCULUS

perpendicular to FV<r, hence we must have for all such
VVt — 0, or a = 0. The latter is obviously out of the
question and hence VVt = 0, that is t = Vw, or we may
choose to write it r = Vu/u.

Hence, VV<r+ VVua/u = 0 = Vv(ua), and S(ua)dp=0
is thus proved to be exact.

We may also proceed thus. Since every vector line is
the intersection of two surfaces, say u = 0 = v, then we can
write the curl of a, which is a vector, in the form

VV<r = hVVu\7v,

and if S<tS7<t = 0, it follows that we must have a in the
plane of Vw, Vfl and

a = xVu -f- yVv. Sadp = — xdu — ydv.

But also

VVcr = VVxVu + VVyVv = hVVuVv.

Hence

SVuVyVv = 0 = SVvVxVu.

These are the Jacobians of u, v, x and u, v, y however, and
since their vanishing is the condition of functional de-
pendence, it follows that x and y are expressible as functions
of u and v. Hence we have

x(u, v)du + y(u, v)dv — 0.

It is known, however, that this equation in two variables is
always integrable by using a multiplier, say g. Therefore
S(ga)dp = 0 is exact for a properly chosen g. Further we
see that ga = — Vw, or that when SaV.a = 0, a = mVw.
If SVo- = 0 for all points, then we find easily that

a = Wr.
For

a = hVVu\/v,

DIFFERENTIALS 193

so that

SV<t = SvhVuVv = 0
and

h = h(u, v).

Integrate h partially as to u, giving

w = fhdu + f(v),
then

Vw = hVu + fvVv, VX/wVv = hVX/uX/v = o\

Set r = wX/v or — v X7w and we have at once a = VX/j.
It is clear that if we draw two successive surfaces W\
and w2 and two successive surfaces Vi and v2, since

m„ Aw , „, Av

T\/w = and T\7v =

Ani An2

and the sides of the parallelogram which is the section of
the tube are A<?2 = Arii esc 6, Asi = An2 esc 6, and
area = AniAn2 esc 6, then TVx area = AwAv, and these
numbers are constant for the successive surfaces, hence the
four surfaces form a tube whose cross-section at every point
is inversely as the intensity of a. For this reason a is said
to be solenoidal or tubular.

If Vx/a = 0 for all points then we must have a = V«.
For SvVo- = 0 and a = gVv, VX/a = VX/gX/v, hence g
is a function of v, and we may write

a = X7u.

If X7d = 0, we must have, since Sx/c = 0, a = VX/r, and
since VX7(T = 0, a = X7u, whence X72u — 0. Therefore, if
X7(T = 0, <t is the gradient of a harmonic function and also the
curl of a vector r, the curl of the curl of r vanishing. Also
if VX7VX/t = 0, since we must then have Vx/t = X/v, and
therefore SV^Vr = 0 = V2fl, we can say that if the curl

194 VECTOR CALCULUS

of the curl of a vector vanishes it must be such that its curl
is the gradient of a harmonic function. Also SdpVr= —dv.
Functions related in the manner of v and r are very im-
portant.
Since in any case SvVVcr — 0, we must have

Vv<r = VVuVv or VV(<r — u\/w) = 0,

whence

a — uVw = Vp,

so that in any case we may break up a vector a into the form

a = Vp + uVw.

It follows that SaV<r = SVp\7u\/w. If we choose u, w
and x as independent variables, we have

Vp = PxVx + puVu + pw Vw,
whence

S<tX7(t = pxSVxVuVw,

and we can find p from the integral

p = fSaVv/SVxVuVw-dx.

In case SaVcr = 0, p = constant, and a = uVw.

A theorem due to Clebsch is useful, namely that a can
always be put into the form

<r = Vp + VVt, where V\/Vp = 0, SVFVr = 0,

that is, <r can always be considered to be due to the super-
position of a solenoidal field upon a lamellar field. We
merely have to choose p as a solution of

V2p = SVcr,

for we have at once Sv(<r — Vp) = 0, and therefore

o- — Vp = VVt.

DIFFERENTIALS 195

This may easily be seen to give us the right to set

<r = Vp + (Vv)nr.

EXAMPLES. SOLUTIONS OF CERTAIN DIFFERENTIAL
FORMS

(1). SV<t = 0, then a = VVr, and if Vv<r = 0, <r = Vp.
If V<r = 0, <7 = Vh where V2^ = 0.

(2). If <p is a linear function dependent upon p continu-
ously, and <pV = 0, <p = OVvQ- If <poV = 0,

<Po = VV(60VV0),

8, do are linear functions. For the notation see next chapter.

(3). VVvQ = 0, <p = - VSaQ. If e(Fv </>()) = 0,
<P = fcFVO ~ V-SerO. If (FV^())o = <W = p() - V^().
Fv^o = 0, <?o = - S()V- Vp.

(4). A particular solution of certain forms is given, as
follows :

*SVo" = a, cr = Jap, Fv<r = eat, a = \Vap,
Vp = oc, p = — Sap, yXJ = ol, (p = — Sap'Q,
VV<pQ = 6, <p = - iVpdQ, €(VV<pQ) - a,

? = - &*p.(), (Fvrio = 0O, ip ;- - i^oO,

Fw = p{), <p - - fo7p() - V&r().

EXERCISES

1. Consider the cases o- = t -\-jf(g(p)) + cfc, where/ and gr have the
following values: f = g, g2, g3, <g,fg, g~\ g~2, e«, log g, sin g, tan #, and
g has the values y/r, (y - 'ax) /(ay + »), (bx + jf)/(a; - &y), x/y,
— x/y, — y/x, etc., V (x2 + y2) — a.

2. Consider the vector lines of

a = i cos (3-n-r) + j sin (3xr), r = V (x2 + y2).

3. Consider the significance of S-Ua\/Ua = 0; give examples.

4. If rf<r = Vt dp find F V <r. Likewise if da = adpd, da = aSpdp, da
= —p2dp, da = Vradp where t is a function of p.

17. Groups. If Si, Sj, • • • , Sn are any functions linear

196 VECTOR CALCULUS

in V but of any degrees in p, then they form a transforma-
tion group (Lie's) if and only if for any two Si, S;,

where 0 is a linear function of Si, S2, • • • Sn, and a, /?
arbitrary vectors. For instance, we have a group in the
six formal coefficients of the two vector operators

Si = - V - pSpV, S2 = - FpV,

for

SaZiSpEi - S0Ei&*Ei = Sa(3Z2,
SaE2S/3E2 - £/3E2&*E2 = - &x/3S2,
&*SiS/3S2 - S/SSt&xSi = - SapBi.

The general condition may be written without a, /3 :

Kt S E/ - Si'SZj - v e 0,

where the accented vector is operated on by the unaccented
one.

Integration
18. Definition. We define the line integral of a function
of p,f(p), by the expression

flf(p)<p{dp) = Lim 2f(Pi)(p(dpi), % - 1, • • •, »j

n = 00

where the vectors pi for the n values of i are drawn from the
origin to n points chosen along the line from A to B along
which the integration is to take place, <p(cr) is a function
which is homogeneous in a and of first degree, rational or
irrational, dpi = pt- — pz_i, and the limit must exist and be
the same value for any method of successive subdivision
of the line which does not leave any interval finite. Like-
wise we define a definite integral over an area by the expres-
sion

ffi(p)<P2{dip,d2p) = Lim 2f(j>i)<to(dipit d2pi),

INTEGRALS 197

where <p2 is a homogeneous function of dipi and d2p{, two
differentials on the surface at the point pi, and of second
degree. A definite integral throughout a volume is simi-
larly defined by

J%J%.ff(p)<P3(dip, d2p, dzp) = Lim 2/(p»)¥>g(dipt, d2pi, d3pi).

For instance, if we consider /(p) = a, we have for ffadp
along the straight line p = fi + #7, dp = cfo-7 and

Lim "Zadx-y from # = #0 to x = Xi is 0:7(21 — Xo),
hence

^P = «(Pi - Po).

The same function along the ellipse p = /3 cos 0 + 7 sin 0,
where dp = (— /? sin 6 + 7 cos 0)d0 has the limit

(a/3 cos 6 -\- ay sin 0)

between 0 = 0O, 6 = 0i, that is, again a(pi — p0).

EXAMPLES
(1). j£« £dp/p = log TWpo, for any path.
(2)- Su ~ q~ldqq~l = qr1 — g0_1, for any path.
(3). The magnetic force at the origin due to an infinite
straight current of direction a and intensity / amperes is
H = 0.2-I-Va/p, where p is the vector perpendicular from
the origin to the line. In case then we have a ribbon whose
right cross-section by a plane through the origin is any
curve, we have the magnetic force due to the ribbon,
expressible as a definite integral,

H = 0.2IfVaTdp/p.

For instance, for a segment of a straight line p = a(3 -\- xy,
/3, 7 unit vectors Tdp = dx,

H = 0.27 / '(ay - xt3)dx/(a2 + x2)
= - 0.2/0 -log (a2 + *22)/(a2 + *i2)

-f- 0.2 -I-yitsoT1 x2/a — tan-1 xj/a),
= 0.27/3 -log OA/OB + O.27J. L AOB.

198 VECTOR CALCULUS

(4). Apply the preceding to the case of a skin current in a
rectangular conductor of long enough length to be prac-
tically infinite, for inside points, and for outside points.

(5). Let the cross-section in (4) be a circle

p — b3 — a(3 cos 6 — ay sin 0.

Study the particular case when b = 0 and the origin is the
center.

(6). The area of a plane curve when the origin is in the
plane is

\TfVpdp.

If the curve is not closed this is the area of the sector
made by drawing vectors to the ends of the curve. If we
calculate the same integral \fVpdp for a curve not in the
plane, or for an origin not in the plane of a curve we will
call the result the areal axis of the path, or circuit. This
term is due to Koenigs (Jour, de Math., (4) 5 (1889), 323).
The projection of this vector on the normal to any plane,
gives the projection of the circuit on the plane.

(7). If a cone is immersed in a uniform pressure field
(hydrostatic) then the resultant pressure upon its surface is
"~ 2^Vpdp-P, where p is taken around the directrix curve.

(8). According to the Newtonian law show that the at-
traction of a straight segment from A to B on a unit point at
0 is in the direction of the bisector of the angle AOB,
and its intensity is 2/x sin ^AOB/c, where c is the perpen-
dicular from 0 to the line.

(9). From the preceding results find the attraction of an
infinite straight wire, thence of an infinite ribbon, and an
infinite prism.

(10). Find the attraction of a cylinder, thence of a solid
cylinder.

19. Integration by Parts. We may integrate by parts

INTEGRALS 199

just as in ordinary problems of calculus. For example,

fysV-adpSpP = iVa(B8P8 - ySfa) + \VaVPf*V pdp,

which is found by integrating by parts and then adding to
both sides J*yV -adpSpp. The integral is thus reduced to
an areal integral. In case y and 5 are equal, we have an
integral around a loop, indicated by J?.

EXAMPLES

(1). SfdpVcxp = HdVaS - yVay) - \Vaf*Vpdp

+ iSafjVpdp.

(2). fy*V.VadpV(3p = ilaSPSy'Vpdp + pS-afy'Vpdp

- 5 Sap 5 + y Softy].

(3). fy*S'VadpV(3p = i(Sa8S(35 - Say Spy - 82SaP

-y2Sap- S-a(3fysVpdp).

(4). JfV-adpVPp = U*SpfysVpdp + pSafjVpdp

- dSa(38 + y Softy + Sa5S(38

- Say Spy - 82SaP + y2SaP

- SaPffVpdp).

(5). fysSapSpdp = USadSpd - SaySPy

-S-Voftf'Vpdpl

(6). ffdpSap = itfSad - ySay + V-affVpdp].

(7). fysVaPSpdp = HVadSpb - VaySPy - SoftffVpdp

+ PSaJfVpdp].

(8). fysVap-dp = i[Va6'B - Vayy + afy8Vpdp

+ SaffVpdp].

(9). fjapdp = h[a(82 - y2) + 2af*Vpdp].

As an example of this formula take the scalar, and notice
that the magnetic induction around a wire carrying a

200 VECTOR CALCULUS

current of value Ta amperes, for a circular path a

B - - 2p.Vap/a2.
Therefore

- fO^Sapdp/a2 = - SfdpB = - OSfia^SafVpdp
= ATafia~2wr2.

For fj, = 1, r = a, this is OAwC. This gives the induction
in gausses per turn.

(10). SfSdpw - i[S8cp8 - Sy<py] + SeffVpdp.
(11). /^prfp = h[Vy<py - V8<p8 + <p'f*Vpdp

+ rmffVpdp]*
(12). XVprfp = }[**.« - ^y.7 + SeffVpdp

- tp'f'Vpdp] - m.ffVpdp.
For any lineolinear form

SfQip, dp) = hm, «) - Q(y, y)]

+ ifAQiP, dp) - Q(dP, p)}
= ««(*, *) - Q(r t)] + WSfVpdp.

(13). State the results for preceding 12 problems for in-
tegration around a loop.

(14). Consider forms of second degree in p, third degree,
etc.

20. Stokes* Theorem. We refer now to problem
page 189, where we have the value of cro, a function of po,
stated for the points in the vicinity of a given fixed point.
If we write <tq for the value of a at a given origin 0, its
value at a point whose vector is dp is

o- = V5p[- S<r08p + %S8pVS(ro8p] - £F5pFVo%

where V refers only to <r0, and gives a value of the curl at

* wii(v) = — Si(pi — Sj<pj — Sk<pk. For notation see Chap. IX.

INTEGRALS 201

the origin 0. If we multiply by ddp and take the scalar,
we have

Sadbp = dSp[Sa08p - iS8pVSa0dp] + iSSpd6pVV<r0.

Therefore if we integrate this along the curve whose vector
radius is dp we have

ffcSed&p = [So-08p2 - Saodpi - §S6p2VS<ro8p2

+ iSSpiV Saotpi] + %SW<T0fVdpd8p.

The last expression, however, is the value of

$[FVovareal axis of the sector between dpi and 5p2].

Therefore for an infinitesimal circuit we have

fSvodbp = £[FVovareal axis of circuit] = SUvVVvo-dA.

FWo is the curl of a at some point inside the loop. If now
we combine several circuits which we obtain by subdividing
any area, we have for the sum of the line integrals on the
left the line integral over the boundary curve of the area
in question, and for the expression on the right the sum of
the different values of the scalar of the curl of a multiplied
into the unit normals of the areas and the areas themselves
or the area integral ffSV\/(rdipd2p. That is, we have
for any finite loop, plane or twisted, the formula

fSadp = ffSVV(TVdlPd2p.

This is called Stokes' Theorem. It is assumed in the proof
above that there are no discontinuities of a or V\/a,
although certain kinds of discontinuities can be present.
The diaphragm which constitutes the area bounded by
the loop is obviously arbitrary, if it is not deformed over
a singularity of a or V\7a.

It follows that fSadp along a given path is independent
of the path when the expression on the right vanishes for
X4

202 VECTOR CALCULUS

the possible loops, that is, is zero independently of dip,
dip, or that is, V\7<r = 0. This condition is necessary and
sufficient.

It follows also that the surface integral of the curl of a
vector over a diaphragm of any kind is equal to the circula-
tion of the vector around the boundary of the diaphragm.
That is, the flux of the curl is the circuitation around the
boundary.

We may generalize the theorem as follows, the expression
on the right can be written ffSUvVVo- dA, where v is
the normal of the surface of the diaphragm and dA is the
area element. If now we construct a sum of any number
of constant vectors au a2, • • • an each multiplied by a
function of the form Saidp, Scr^dp, • • • Scrndp, we will have
a general rational linear vector function of dp, say <pdp,
and arrive at the integral formula

fvdp = ff<p(VUpV)dA,

where the V refers now to the functions of p implied in <p.
This is the vector generalized form of Stokes' theorem.

If the surface is plane, Uv is a constant, say a, so that
for plane paths

fipdp = ff<pVVa-dA.

We may arrive at some interesting theorems by assigning
various values to the function <p. For instance, let

<pdp = a dp,
then

<p(VUvV) = <t'VUvVv'=-Ui>SV<t+V'S<t'Uv+SUpV(t,

whence

ffS^a-dv = ffV'Sa'dv+ fVadp.
If

<fdp = pSdpa,

INTEGRALS 203

then

<pVUvV = pSUvVo- - VaUv,
therefore

ffV-adv = ffpSdvVa - fpSadp.
If

ipdp = pVdpa,
tpVUvV = pV(VUvV)<r - SUva + aVv,

therefore

ffvdv + Sadv = - ffpV(VUvV)<r + fpVdpa,

hence

2ffSadv = - ffSp(VUvV)<r + fSpdpa.

EXERCISES

1. Investigate the problems of article 19, page 198, as to the applica-
tion of the theorem.

2. Show that the theorem can be made to apply to a line which is
not a loop by joining its ends to the origin, and after applying the
theorem to the loop, subtracting the integrals along the radii from 0
to the ends of the line, which can be expressed in terms of dx, along a line.
Also consider cases in which the paths follow the characteristic lines of
Vadp = 0.

3. The theorem may be stated thus: the circulation around a path
is the total normal flux of the curl of the vector function a through the
loop.

4. If the constant current la amperes flows in an infinite straight
circuit the magnetic force H at the point p (origin on the axis) is for

Tp<a H = ^IVaP,
and for

a<TP H = 0.2a?I/VaP,

a is the radius of the wire. Then 7vH = /(a/10) inside the wire and
equals zero outside. Integrate H around various paths and apply
Stokes' theorem. In this case the current is a vortex field of intensity
7ra27/10.

5. If we consider a series of loops each of which surrounds a given
tube of vortex lines, it is clear that the circulation around such tube
is everywhere the same. If the vector <r defines a velocity field
which has a curl, the elementary volumes or particles are rotating, as

204 VECTOR CALCULUS

we have seen before, the instantaneous axis of rotation being the unit
of the curl, and the vector lines of the curl may be compared to wires
on which rotating beads are strung. It is known that in a perfect
fluid whose density is either constant or a function of the pressure only,
and subject to forces having a monodromic potential, the circulation in
any circuit through particles moving with the fluid is constant. [Lamb,
Hydrodynamics, p. 194.] Hence the vortex tubes moving with the fluid
(enclosing in a given section the same particles), however small in cross-
section, give the same integral of the curl. It follows by passing to an
elementary tube that the vortex lines, that is, the lines of curl, move
with the fluid, just as if the beads above were to remain always on the
same wire, however turbulent the motion. In case the vortex lines
return into themselves forming a vortex ring, this leads to the theorem
in hydrodynamics that a vortex ring in a perfect fluid is indestructible.
It is proved, too, that the same particles always stay in a vortex tube.

6. Show that for a- = a(3S2otP - 2SpP) + £(4#3/Sp - 2SaP), where
Sa& = 0, the integral from the origin to 2a -J- 2/3 is independent of the
path. Calculate it for a straight line and for a parabola.

7. The magnetic intensity H, at the point 0, from which the vector
p is drawn to a filament of wire carrying an infinite straight current in
the direction a, of intensity I amperes, is given by

H = 0.27/Fap.

Suppose that we have a conductor of any cross-section considered as
made up of filaments, find the total magnetic force at 0 due to all
the filaments. Notice that

H = 0.2/ Fa V log TVap,

and that a is the unit normal of the plane cross-section of the conductor.
Hence

ffHdA = ff0.2IVaV log TVapdA = f0.2I log TVapdp

around the boundary of the cross-section. This can easily be reduced
to the ordinary form 0.21 j? log rdp. This expression is called a log-
arithmic potential. If I were a function of the position of the filament
in the cross-section, the form of the line-integral would change.

For a circular section we have the results used in problem 4. Con-
sider also a rectangular bar, for inside points and also for outside points.

8. If or and r are two vector functions of p, we have the theorem

SVUuVVo-t = St(VUvV)* - S<t(VUpV)t,
whence

ffSr(VUpV)o- = ffS<r{VUvV)r + fSdpar,

INTEGRALS 205

for a closed circuit. Show applications when a or t or both are sole-
noidal.

9. Show that

ffS-dvotS\7<x = fSdpaa + ffSdv(SaV)<r,
ffS-Vuadv = JTuSadp - f fuSV adp,
ffS-X7uS7vdv= fuSsjvdp = - fvS\7udP,
f1hiSVvdP= [uv]ppl - f^vSS/udp.

10. Prove Koenig's theorems and generalize.

(1) Any area bounded by a loop generates by translation a volume
= — Saw, where co is the areal axis;

(2) The area for a rotation given by (a + Vap)at is — J] Saco +
ftScf VpVpdp.

21. Green's Theorem. The following theorem becomes
fundamental in the treatment of surface integrals. Refer-
ring to the second form in example 17, page 189, for the
expression of a vector in the vicinity of a point, which is

0" = FVSp[- iV8pa0 + iS8pVVdp<ro] - l&pSV<To

we see that if we multiply by a vector element of surface,
Vdi8pd2dp, and take the scalar

Scdrfpdidp = SUvVsp[]dA - iSV(r0Sdldpd2dp8p.

If now we integrate over any closed surface the first term
on the right gives zero, since the bounding curve has be-
come a mere point, and thus, indicating integration over
a closed surface by two J',

j> $&<jd\hpd<ihp = — \S\7(TQjf jfSdibpd'ibpbp.

But the last part of the right hand member is the volume
of an elementary triangular pyramid whose base is given by
didpd28p. Hence, the integral is the elementary volume of
the closed surface, and may be written dv, so that we have
for an elementary closed surface

j> \fSad18pd2dp = SVvodv.

206 VECTOR CALCULUS

If now we can dissect any volume into elements in which
the function has no singularities and sum the entire figure,
then pass to the limit as usual, we have the important
theorem

ffS<rdlPd2p = fffSVv dv.

This is called Greens theorem, or sometimes Green's theorem
in the first form. It is usually called Gauss' theorem by
German writers, although Gauss' theorem proper was only
a particular case and Green's publication antedates Gauss'
by several years.

The theorem may be stated thus: the convergence of a
vector throughout a given volume is the flux through the
bounding surface.

It is evident that we can generalize this theorem as we did
Stokes' and thus arrive at the generalized Green's theorem
$ fQvdA = f f f$\/ dv. v is the outward unit normal.

The applications are so numerous and so important that
they will occupy a considerable space.

• The elementary areas and volumes used in proving Stokes'
and Green's theorems are often used as integral definitions
of convergence or its negative, the divergence, and of curl,
rotation, or vortex. For such methods of approach see Joly,
Burali-Forti and Marcolongo, and various German texts.

A very obvious corollary is that if SVc = 0 then

$ \fSad1pd2p = 0.

It follows that the flux of any curl through any closed sur-
face is zero. Hence, if the particles of a vortex enter a
closed boundary, they must leave it. Therefore, vortex
tubes must be either closed or terminate on the boundary
wall of the medium in which the vortex is, or else wind
about infinitely. We may also state that if SVa = 0 the
differential expression Sadipd2p is exact in the sense that

INTEGRALS 207

J%J%S(rdipd2p is invariant for different diaphragms bounded
by a closed curve, noting the usual restrictions due to
singularities.

We proceed to develop some theorems that follow from
Green's theorem. Let $Uv be — pSUvcr, then

3>V = — pSv<r + o-
and we have

fffadv = fffpSVvdv - ffpSUvadA.

Let $Uv = — pVUva, then <i>V = — pVVv + 2a and
SSfvdv = ifffpVVvdv - \ffpVUvodA.

Let $Uv = pSpUva, then <J>V = pSpVv + Fpo-, whence
fffVpa dv = - fffp&Va dv + ffpSpUvadA.

Let $17V = - pVpVUixr, then $V = - pFpFVo" + 3PV,

hence

SSSVpadv = ifffpVpWadv- \ffpVpVUvadA.

Let $E7V = SprUiHT, then 3>V = SprV<r + Spa\/r + Sot,

thence

fffSar dv = - fffiSprV* + &rVr)<fo

+ f f&prTJva dA.

In the first of these if a- has no convergence we have the
theorem that the integral of cr, a solenoidal vector, through-
out a volume is equal to the integral over the surface of p
multiplied by the normal component of a. In the second
we have the theorem that if the curl of a vanishes through-
out a volume, so that a- is lamellar in the volume, then the
integral of a throughout the volume is half the integral
over the surface of p times the tangential component of a
taken at right angles to a-. In the third, if the curl of cr

208 VECTOR CALCULUS

vanishes then the integral of the moment of a with regard
to the origin is the integral over the surface of Tp2 times the
component along p of the negative of the tangential com-
ponent of a taken perpendicular to <r, and by the fourth
this also equals the surface integral of the component
perpendicular to p of the negative tangential component of
<r taken perpendicular to a. According to the fifth formula,
if a solenoidal vector is multiplied by another and the scalar
of the product is integrated throughout a volume, then the
integral is the integral of — SpaVr throughout the volume
-f- the integral of ScrprUv over the surface.

If in the first, second, third, and fourth we set c<t for a,
and in the fifth ca for a and — \<t for r, we have from the
first and second the expression for X, the momentum of a
moving mass of continuous medium, of density c, and from
the third and fourth the moment of momentum, /x, and
from the fifth the kinetic energy. If the medium is in-
compressible, and we set 2k = V\/v, since SVca = 0, then

X = fffcadv
= - ffcpSUvadA + fffpSaVcdv

+ SSfcpSV* dv

= fffpcKdv+lfffpWcadv - \££cpVVvadA.

ju = fffcVpadv
= ffcpSpUvadA - SSfcpSpVa - fffpSpVca
= UffcpVpK + \fffpVpWcadv

- \ffcpVpVVvadA.
T = - hSfSSa2cdv

= - hffSpvUvac dA + SffhcSpaVo- dv

+ hfffSpvVca dv.

In case c is uniform these become still simpler.

If we set a = S/u and r = \/w in the above formula we

INTEGRALS 209

arrive at others for the gradients of scalar functions. The
curls will vanish. If further we suppose that u, or w, or
both, are harmonic so that the convergences also vanish
we have a number of useful theorems.

Othei forms of Green's theorem are found by the follow-
ing methods. Set $Uv = uS\7wUv, then

$V = u\/2w + SVuVw

and we have the second form of Green's theorem at once

SfS&VuVw dv = ffuS\/wUv dA — fffu\72wdv,

and from symmetry

yWSvWw dv = ffwSVuUv dA — fffw\/2u dv.

Subtracting we have

J %J *J %(u\72w — w\72u) dv

= ~ f£(,STJv[u\7w - wVu])dA.

22. Applications. In the first of these let u = 1, then
fffV2™ dv — — J'.fSUj'VwdA. If then w is a har-
monic function, the surface integral will vanish, and if V2w
= 47Tju, which is Poisson's equation for potentials of forces
varying as the inverse square of the distance, inside the
masses, ju being the density of the distribution, then

ffSUvS7w dA = ±ttM,

where M is the total mass of the volume distribution. This
is Gauss' theorem, a particular case of Green's. In words,
the surface integral of the normal component of the force
is — 47r times the enclosed mass. The total mass is l/4x
times the volume integral of the concentration.

In the first formula let u = 1/Tp and exclude the origin

210 VECTOR CALCULUS

(a point of discontinuity) by a small sphere, then we have

fffSV(l/Tp)Vwdv

= ffdA SUrVw/TP - fffdv V2w/Tp

for the space between the sphere and the bounding surface
of the distribution w, and over the two surfaces, the normals
pointing out of the enclosed space. But for a sphere we
have dA = Tp2 dw where co is the solid angle at the center,
and dv = Tp2dwdTp. Thus we have

fffV2w/Tp dv

= ffSdA UuVw/Tp -fffSv(l/TP)Vwdv
= ffSdA UvVw/Tp -fffSv(wV[l/TP])dv

since V2l/7p = 0,
= ffSdA UvVwjTp -ffSdA UvwV(l/Tp)
= ffSdA UvVwjTp+ffSdA VvVp\T2pw.

Now of the integrals on the right let us consider first the
surface of the sphere, of small radius Tp. The first integral
is then - ffSUpX/wlTp- T2pdco = - ffSUpVw- Tpda,
and if we suppose that the normal component of Vw, that
is, the component of Vw along p, is everywhere finite, then
this integral will vanish with Tp. The second integral for
the sphere is — J?rf'SUpUpwT2pd(x)lT2p = — tfj'wdu,
and the value of w at the origin is Wo, then this integral is
47TWo since the total solid angle around a point is 47r.
Hence we have

fffdv V2w/Tp = ffSUv{\/wlTp + wUp/T2p)dA

+ 4twq
and

4x^o= fffdvV2w/Tp

- ffSUp(Vw/Tp -f wUp/T2p) dA,

INTEGRALS 211

where the volume integral is over all the space at which w
exists, the origin excluded, and the surface integral is over
the bounding surface or surfaces. In words, if we know the
value of the concentration of w at every point of space,
and the value of the gradient of w and of w at every point
of the bounding surfaces at which there is discontinuity,
then we can find w itself at every point of space, provided
w is finite with its gradient. If X72w is of order in p not
lower than — 1 we do not need to exclude the origin, for
the integral is ///V2^ TpdcpdTp, and this will vanish
with Tp when V2w is not lower in degree than — 1.

EXERCISES

1. We shall examine in detail the problem of w — const, over a given
surface, zero over the infinite sphere, V2w = 0 everywhere, \/w = 0
on the inside of the sphere, but not zero on the outside. Then for the
inside of the sphere the equation reduces to

4:irw0 = - £fwSUvUplT*pdA = 4ttu;,

hence w is constant throughout the sphere and equal to the surface value.
On the outside of the sphere, we have to consider the bounding sur-
faces to be the sphere and the sphere of infinite radius, so that we have

4^0 = _ ffSdA UvVw/Tp- wffSdA UuUpfTp2,

where the first integral is taken over both surfaces and the second
integral is over the given surface only, since w = 0 at °° . The second
integral vanishes, however, since it is equal to w times the solid angle
of the closed surface at a point exterior to it. If we suppose then that
\/w is 0 at «3 we have a single integral to evaluate

4:irw0 = — j> j> 'SdAU ri>\? 'w/T 'p over the surface.

A simple case is

— SUv\/w = const. = C.
Then

4ttWo = CffdAITp.

The integration of this and of the forms arising from a different assump-
tion as to the normal component of V^ can be effected by the use of
fundamental functions proper to the problem and determined by the
boundary conditions, such as Fourier's series, spherical harmonics,
and the like. One very simple case is that of the sphere. If we take

212 VECTOR CALCULUS

the origin at the center of the sphere we have to find the integral

,f,fdA/T(P - Po)

where po is the vector to the point. Now the solid angle subtended
by po is given by the integral — r~lffdASpU{p — po)/T*(p — p0)
and equals 4t or 0, according as the point is inside or outside of the
sphere. This integral, however, breaks up easily into two over the
surface, the integrands being

r-^T-Kp - po) - SpoU(P - P0)/T*(p - po),

but the last term gives 0 or — 47rr2/7Tp0, as the point is inside or outside
of the sphere. Hence the other term gives

ffdAlT{p - po) - 47rr or 4Trr2/Tp0

as the point is inside or outside. We find then in this case that

w0 m Cr2/Tpo.

If in place of the law above for — SUvS7w, it is equal to C/T2(p — p0)
we find that

ffdAIT\P - po) = 47rr/(r' + p02)
or

47^/(7^0 - r^po).
Inside

_ r r,A S(p ~ pp)(p + po)

- ffdA TKp - po) '

dA = 27rr2 sin Odd =- d[a2 + r2 - x2] = —xdx,
a a

„po(p — po) = ax cos 4/ _ a2 + x2 — r2
T*(p - po) " x3 2x*

ffdAS^f^=^fr+a'a+r(a^ + l)dx = 0

T2(p — po) aJr-a,a-r \ X2 J

or

47TT2

a

The differentiation of these integrals by using Vp0 as operator under
the sign leads to some vector integrals over the surface of the sphere.
2. Show that we have

££UvdAIT(p - po) = |ttpo or |7rr3/^3Po-po

for inside or outside points of a sphere.

INTEGRALS 213

3. Find ffdAUu/T3(P - Po) for the sphere.

4. Prove f fdAT^{p-fi)T-\p-oc) =47rr/[(r2-a2)77(/S-«)] or

= ^r2J[a(r2-a2)T(r2a-1 +0)].

5. Consider the case in which the value of w is zero on a surface
not at infinity but surrounding the first given surface. We have an
example in two concentric spheres which form a condenser. On the
inner sphere let w be const. = Wi, on the outer let w = 0, on the inner
let — SUpVw = 0, inside, = Eh outside, on the outer let — SUv\/w
= E2 on the inside, = Oon the outside.

6. If w is considered with regard to one of its level surfaces, it is
constant on the surface, and the integral — £ f SdAU vU p\T2 pio
becomes for any inside point 4:irw, hence we have

4irw0 - A.™ = fffdv\72wlTP - £ £SdAUuVw/TP.

If then w is harmonic inside the level surface, it is constant at all points
and

47r(w0 - to) m - £fSdAUv\7wlTp.

But since w0 is constant as we approach the surface, V^o =0, and
V(w — Wo) = 0, so that X7w = 0. Hence w0 = w. If w vanishes at
oo and is everywhere harmonic it equals zero.

7. If two functions Wi, w2 are harmonic without a given surface,
vanish at » , and have on the surface values which are constantly in the
ratio X : 1, X a constant, then W\ = \W2.

8. If the surface Si is a level for both the functions u and w, as also
the surface S2 inside Si, and if between Si and $2, u and w are harmonic,
then

(U — Ui)(w2 — Wi) = (W — Wi)(ll2 — Ui).

For if w = <p(u), then V2w = 0 = <p"(u)T2\7u, hence <p(u) — au + b,
etc.

[A scalar point function w is expressible as a function of another
scalar function u if and only if V\/w\7u = 0.]

9. Outside a closed surface S, Wi and w2 are harmonic and have the
same levels. Si vanishes at • while w2 has at 00 everywhere the con-
stant value C. Then w2 = Bwi + C.

For Vw2 = tVwh V2w2 = V^V^i = 0, thus V* = 0, or V^i = 0,
and t = B or wi = const.

10. There cannot be two different functions W\, w2 both of which
within a given closed surface are harmonic, are continuous with their
gradients, are either equal at every point of S or else SUvX/Wi =SUv\/w2
at every point of S while at one point they are equal.

Let u = Wi — w2, then V2w = 0, SJu = 0 on S or else SUv\/u = 0,
and at one point Vw = 0.

214 VECTOR CALCULUS

11. Given a set of mutually exclusive surfaces, then there cannot be
two unequal functions w\, Wi, which outside all these surfaces are
harmonic, continuous with their gradients, vanish at <» as Tp~l, their
gradients vanishing as Tp-2, and at every point of the surfaces either
equal or SUvVwi = SUvVwt-

23. Solution of Laplace's Equation. The last problems
in the preceding application show that if we wish to invert
V2w = 0, all we need are the boundary conditions, in order
to have a unique solution. In case V2u is a function of
P>f(p)> we can proceed by the method of integral equations
to arrive at the integral. However the integral is express-
ible in the form of a definite integral, as well as a series,

w0 = l/4:w[fSSdvV2w/Tp

- ffSUviVw/Tp + wUp/T2p)dAl

The first of these integrals is called the potential and written
Pot. Thus for any function of p whatever we have

Vot q, = fffqdvlT(p- pQ)

where p describes the volume and p0 is the point for which
Pot qo is desired. Let Vo be used to indicate operation as
to po, then we have

Vo Pot g0 = VoffSqdv/T(p - p0)

= fff[dvU(p - p0)/r2(p - Po)]q
- -SSfV[qlT(p- p0)]dv

+ SffdWq/T(p- po)
= Pot Vg - ffdAUvqlT(p - Po).

If we operate by Vo again, we have

Vo2 Pot q0 = Pot V2? - ffdA[Uv\7qlT(p - po)

+ V'Uvq/T'(p - po)].

But the expression on the right is 4x^0, hence we have the

INTEGRALS 215

important theorem

Vo2 Pot q0 = 4:irq0.

That is, the concentration of a potential is 4x times the
function of which we have the potential. In the case of a
material distribution of attracting matter, this is Poisson's
equation, stating that the concentration of the potential
of the density is 4r times the density; that is, given a
distribution of attracting masses, they have a potential at
any given point, and the concentration of this potential at
that point is the density at the point -5- 4-7T.

The gradient of Pot q0 was called by Gibbs the Newtonian
of g0, when the function q0 is a scalar, and if q0 is a vector,
then the curl of its potential is called the Laplacian, and the
convergence of its potential is called the Maxwellian of q0.
Thus

New q0 = Vo Pot P, Lap <r0 = V\/o Pot <r0,
Max (To = £ Vo Pot co.

We have the general inversion formula

47rVo~2Vo2? = 47rgo

- SSfV2q/T(p - Po)dv

- ffdA[UvTqlT{p - p0)

+ U(p - p0)qUr/T*(p - p0)J.

This gives us the inverse of the concentration as a potential,
plus certain functions arising from the boundary conditions.
We may also define an integral, sometimes useful, called
the Helmholtzian,

Him. Q m fffQT{p - Po)dv.

Certain double triple integrals have been defined:

Pot 0, v) = ffffffu(p1)v(p2)dv1dv2/T(p1 - p2),

216 VECTOR CALCULUS

Pot (to) = fffffS - Sh dvidvJTfa - p2),
Lap (to) = ffffff + 5to(Pi - p2)^i^2/P(Pl-p2),
New («, f) = SSSSSS-S{i(pi-p2)vldvidv2IT'(pi-pt),
Max(£,*) = - ffffffvlSUpi-P2)dvldv2ir(Pl-P2).

EXERCISES

1. Iff = — VP is a field of force or velocity or other vector arising
from a scalar function P as its gradient, then

Po = - SSSSV£dv/[4irT(P- po)] + ffdA[SUvll&*T{p - po))

+ PC/yV^(p-po)/47r].

If P is harmonic the first term vanishes, if £ = 0 the first two vanish.

2. If £ = V<r, that is, it is a curl of a solenoidal vector,

°o = fffVV* dv/[4irT(p - po)] - f<fdA[VUv<rl[±TcT{p - po)]

+ U(p - p9)<rlU,[4*T*(j> - po)].

3. We may, therefore, break up (in an infinity of ways) any vector
into two parts, one solenoidal and the other lamellar.

Thus, let a = 7T + t where £v r ■ 0, and Wir = 0, then Sv <r = SVx
and since VVt = 0, this may be written Vt = &Vo" whence x.

VVc = FVt = Vr whence t. We have, therefore, from these two
47T<r0 = VfffSdvV<r/T(p - Po) - V £ £SdAUvalT{p - Po)

+ V jffPSdA UvV (UT(p - po) + V V SSfVV*dv/T(p - Po)
-WffVU*adA/T(p-po) + VVffDSU*S7(p+(P-po)dAt

where P is such that V2P = *S'V<r and D such that \72D = Vs/a.

3. Another application is found in the second form of Green's
theorem. According to the formula

SffiyW1™ - wV2u) dv = - tf£(SUv[u\7w - w\/u])dA

it is evident that if G is a function such that V2G = 0, and if, further,
G has been chosen so as to satisfy the boundary condition SUvS7G = 0,
then the formula becomes

SffGs^wdv = - ££SUv\7wGdA.

If then V2w is a given function we have the integral equation

JfGSUvVwdA - - fffGj{P)dv.

Similar considerations enable us to handle other problems.

4. If u and w both satisfy V2/ = 0, then we have Green's Reciprocal
Theorem:

ffuSUvSfw dA = ffwSUvVudA,

Thus let
therefore

INTEGRALS 217

ff ^p dA = ffuSUvV -L dA.

5. Let A relate to a as V to p; then

A Pot Q = ff/QdvU(p - a)/T*(P - a)

= fffV(Q/T(p - cc))dv + fffdWQ/T(p - a)

= Pot VQ - ffdAUuQ/T(P - a).

If Q — 0 on the surface, the surface integral = 0.

New P = Pot V - ffVvP dAjT{p - a) = A Pot P when Pot exists.

Lap a = V Pot Vo- - ££VTJvadA\T{p - a) = VA Pot a when Pot
exists.

Max o- = S Pot V<r - £ fSUvadAITip -a) = SA Pot o- when Pot
exists.

A2 PotQ = Pot V2Q - ££Uv\7QdAIT(p - a)

+ //diVi[^/77i(P -«)].
If Q = 0 on the surface, that is, if Q has no surface of discontinuity,

A2 Pot Q - Pot V2Q,

A New P = A2 Pot P,

A Lap o- = A7A Pot a,

A Max a = A/SA Pot <r.

6. If j8 is a function of the time t, then

d--^[yy/i(rvFv^+M^J

t+br

r r

+ VV £<fj Vdu pt+br - ff Vdv WPt+br

where the subscript means t + br is put for t after the operations on 0
have occurred.

15

CHAPTER IX

THE LINEAR VECTOR FUNCTION

1. Definition. If there is a vector a which is an integral
rational function <p of the vector p,

a = <p'P,

and if in this function we substitute for p a scalar multiple
tp of p, then we call the vector function a linear vector func-
tion if a becomes ta under this substitution. It is also called
a dyadic.

The function <p transforms the vector p, which may be
in any direction, into the vector <r, which may not in every
case be able to take all directions. If p = a, then we have
(pp = <pa, and <p as an operator has a value at every point
in space. We may, in fact, look upon <p as a space trans-
formation that deforms space by a shift in its points leaving
invariant the origin and the surface at infinity. In the
case of a straight line

Vap = /?, or p = xa + cTl(5,

we see that the operation of <p on all its vectors gives

a = x<pa + (pVa~1^f

and this is a straight line whose equation is

Vipaa = V<pa(pVa~1^,

which will later be shown to reduce to a function of (3
only, <p(3. Hence <p converts straight lines into straight
lines. The lines a for which Vacpa = 0, remain parallel
218

THE LINEAR VECTOR FUNCTION 219

to their original direction, others change direction. Again
if we consider the plane S-afip = 0 or

p = xa + y(3, , a = x<pa + y<pf},

so that

S(r<pct<pp = 0.

Hence planes through the origin, and likewise all planes,
are converted into planes. These will be parallel to their
original direction if Va(3 = uV<pa<p(3, or

VVa$V(pa<p& = 0 = Scx<pa<p(3= S(3<pa<p(3= Sa(3<pa = So@<pP.

Now Va(3 is normal to the plane, and /3 is any vector in the
plane, and <p(3 by the equation is normal to Vafi, hence
<p(3 = va + w(3 for all vectors 0 in the plane.

Since <p0 = 0, the function leaves the origin invariant.
Consequently the lines and planes through the origin that
remain parallel to themselves are invariant as lines and
planes. These lines we will call the invariant lines of <p,
and the planes the invariant planes of tp.

2. Invariant Lines. In order to ascertain what lines are
invariant we solve the equation

Va<pa = 0, or (pa = ga,

that is

(tp - g)a = 0.

First we write a in the form

aS\fiv = \SfJiva + ixSvka + vSXfxa,

where X, ju, v are any three noncoplanar vectors. Then we
have at once

(<p — g)\Sixvicx + (<p — g)nSv\a + (<p — g)pS\fxa = 0.

220 VECTOR CALCULUS

But this means that we must have for any three non-
coplanar vectors X, /i, v

S(<p - g)\(<p - g)fi(<p - g)v = 0

= tfSXiiv — g2(S\ii<pv + S\<ptxi> + S<p\nv)

+ g(S\(pfJL(pV + S\jJl<pV + S(f\(pfJLP) — S<p\<piJ.<pi>,

an equation to determine g, which we shall write

gz - mig2 + m2g - m3 = 0,
called the /a<6n< equation of #>, where we have set

Wl = (S\jA<pV + S\<pflP + S<p\fAl>)/S\fJLl>,

rri2 = (S\(pii(pp -+- S<pkyupv + S(p\<piJLp)lS\fjLv,

These expressions are called the nonrotational scalar in-
variants of <p. That they are invariant is easily seen by
substituting X' + v/jl for X. The resulting form is precisely
the same for Xr, ju, p, and from the symmetry involved this
means that for X, /x, v we can substitute any other three
noncoplanar vectors, and arrive at the same values for
mi, m2, m3. It is obvious that m3 is the ratio in which the
volume of the parallelepiped X, jjl, v is altered. If m3 = 0
one or more of the roots of the cubic are zero. The number
of zero roots is called the vacuity of (p. If is obvious that
the latent cubic has either one or three real roots.

3. General Equation. We prove now a fundamental
equation due to Hamilton. Starting with <p we iterate the
function on any vector, as p, writing the successive results
thus

p, <pp, <p<pp = <p2p, <p<p<pp = <p<p2p = <p3p, "•.

We have then for any three vectors X, ,u, v that are not
coplanar

THE LINEAR VECTOR FUNCTION 221

S\pi>(<p3p — mi<p2p) = (p2(<pp — m\p)S\pv

= <p2[<p\Spvp -\- • - - — pSpv(p\ — • • •]

= - <p2[V-VppV<p\p+ •••]

= <p\V'V<p\pVp.v+ •••]

= <p[<p\Sv<ppp + <pp<S\<pvp + <pvSp<p\p

— <p\SpL(pvp — (fpSvcpXp

— <pi>S\(pfxp].
Adding to this result S\pu> -m%ipp, we have

S\pv((p3p — mnp2p + m<npp)

= <p[\S<pfi<pvp + pSipVipkp + vS<p\(pp,p] = pS(f\cppapv.

Subtracting SXfMV-rritp from both sides and dropping the
nonvanishing factor S\p,i>, we have the Hamilton cubic for <p

<psp — mi<p2p + m*<pp — mzp = 0.

This cubic holds for all vectors p, and hence, may be written
symbolically

<p3 — mnp2 + m2(p — ra3 = 0

identically. This is also called the general equation for <p.
It is the same equation so far as form goes as the latent
equation. Hence we may write it in the form

(<p - gi)(<p — g*)(<p — gz) = 0.

In other words, the successive application of these three
operators to any vector will identically annul it.

We scarcely need to mention that the three operators
written here are commutative and associative, since this
follows at once from the definition of linear vector operator,
and of its powers.

It is to be noted, too, that <p may satisfy an equation of
lower degree. This, in case there is one, will be called the
characteristic equation of <p. Since <p must satisfy its general

222 VECTOR CALCULUS

equation, the process of highest common divisor applied
to the two will give us an equation which <p satisfies also,
and as this cannot by hypothesis be lower than the char-
acteristic equation in degree and must divide it, it is the
characteristic equation. Hence the factors of the char-
acteristic equation are included among those of the general
equation. We proceed now to prove that the general
equation can have no factors different from the factors
of the characteristic equation.

(1) Let the characteristic equation be

(<p - g)p = 0

for every vector; then assuming any X, /x, v, we find easily
for the latent equation

x*-Sgx2+3g2x-g*= 0,

so that the general equation is

(cp - gf = 0.
In this case

if = [g\SM) + gpSrkQ + gpSlnOV&V,

where X, /z, v are given for a given <p.

(2) Let the characteristic equation be

(<P - 9i)(<P - 92) P = 0,

then by hypothesis, there is at least one vector a for which
we have

(<p - gi)a + 0,

and at least one fi for which

(<p - gt)0 4= 0.
Let us take then

O - gi)a = X, (<p— g2)(S = M-

THE LINEAR VECTOR FUNCTION 223

Then

(<p - g2)\ = 0, (<p- gi)fx = 0.

Hence, we cannot have X and ju parallel, else gi = g2, which
we assume is not the case, since from

(<p- g2)U\ = 0, (<p- g1)Ufx= 0,
we have

g2U\ = giUn, and g2 = gu

if X is parallel to /z, that is if U\ would = Up,

There is still a third direction independent of X and /z,
say v. Let

cpv = av + bjjL + cX.
Then we have

(<p - ft)* = (a - gi)j>+ bfx + cX.
Since

(<p- fc)(* - 9i)v = 0,
(a - gx)(<p - g2)v — b(g2 - fi)p = 0

= (a— gY){a — g2)v + 6 (a - g2)fx + c(a — g{)\.

We must have, therefore, either

a = gi and 6=0,
or

a = g2 and c = 0.

As the numbering of the roots is immaterial, let us take
a = git b = 0, then

<pv = giv + cX, <pX = #2X, ^>m = 9iV>

We notice that if c # 0, we can choose v' = v — (cjg2)\,
whence ipv' = giv' and we could therefore take c = 0.
Hence

g3 - g\2gi + g2) + ^(2fir^2 + gY2) - g?g2 = 0,
<p = [guiS\vQ + givSXpQ + #2X£mK)]ASX/xj>,

221 VECTOR CALCULUS

and the general equation is

(<P - 9i)2(<P ~ 92) = 0.
(3) Let the characteristic equation be
(<p - g)*p = 0.
Then there is one direction X for which we have

<p\ = g\,

and there may be other directions for which the same is
true. There is at least one direction \i such that

(cp - g)fi = X.
We have, therefore,

<PV = g» + X <?X = g\.
Let now v be a third independent direction, then we have

(pv = av + bjj. + cK,
(<p - g)v = (a — g) v + 6/x + c\,
(<p - gfv = 0 = (a - gfv + b(a - g)p + [b + c(a - g)]K.

Therefore, we have a = g, 6=0, <pp = gv + cX and
<£>(*> — c/x) = g{y — c/jl) = gv' , and the general equation

i* - g)3 = 0,

<p = g + XSj/XO/SX/x*'.

We are now in a position to say that the general equation
has exactly the same factors as the characteristic equation.
Further we can state as a theorem the following:

(a) // the characteristic equation is of first degree,

O - gY)p = 0,

then every vector is converted into g\ times that vector, by the
operation of (p.

THE LINEAR VECTOR FUNCTION 225

(6) // the characteristic equation is of the form
O - 9i)(<P ~ 92) = 0,

then there is one direction X such that <pk = 92K, while for
every vector in a given plane of the form x\i.-\- yv we have

(<p- #i)Om + yv) = 0.

Hence <p multiplies by gi every vector in the plane of /a, v,
and by g2 all vectors in the direction X.
(c) If the characteristic equation is

W - g,f = 0,

there is a direction such that

<p\ = gi\

and a given plane such that for every vector in it x\x-\- yv
we have

(<P — 9i)(w + yv) = ^X.

If (<P — gi)v = v\ (<P — gi)v = w\ we may set

•
w

giving (p/i = gip. Therefore <p extends all vectors in the ratio
gi, and shears all components parallel to v in the direction X.
4. Nondegenerate Equations. We have left to consider
the three cases

(<p — 9i)(<p — 92) (<p - gz) = 0,

O - gi)2(<P - 02) = 0,

(v - g,f = 0.

In the last case we see easily that there is a set of unit
vectors X, ju, v such that

226 VECTOR CALCULUS

<p\ — g{K + mo,

<PH = giii + vb,

<pv = giv.
Hence we see that

<p(x\ + yi* + zv) = gi{x\ + y\x + zv) + a*M + 6y*>
= gi(x\ + 2/M + •*) + a(a*M + 0*0

+ (6 - a)yy,
<p(x» + yv) = gi(xn + ?/*>) + fo^,

<p = gi + [apSuvQ + bvSv\Q]/S\nv.

Therefore <p extends all vectors in the ratio g\, shears all
vectors X in the direction of m> and all vectors /x in the
direction v.

In the first case we see that there is at least one vector
p such that

{<P - 9\){<P - 9s) P = A,
where

<p\ = g{K.

Likewise there are vectors that lead to /x and v where
<PH = g2n, <pp = gzv. These are independent, and there-
fore if we consider any vector

p = x\ + yn + zp,
we have

<pp = xg{K + 2/02M + zg3v,
<p = [g^SfivQ + fwJSrhO + gsvS\nO]lS\fiP.

Evidently we can find X, /x, v by operating on all vectors
necessary in order to arrive at nonvanishing results by

(<P — 9z)(<P — 9s), (<P — 9i)(<P — 9*)> (<P — 9\)(<P — 9t)

respectively.

In the second case, we see in a similar manner that there

THE LINEAR VECTOR FUNCTION 227

are three vectors such that

£>X = g{K + \x, <pfi = giii, <pv = g2p,
<P = IgiQiSpvQ + nSvkQ + fiwStoQ + jtSMW&Vv.

5. Summary. We may now summarize these results in
the following theorem, which is of highest importance.

Every linear vector function satisfies a general cubic, and
may also satisfy an equation of lower degree called the char-
acteristic equation. If the equation of lowest degree is the
cubic, then it may have three distinct latent roots, in which
case there corresponds to each root a distinct invariant line
through the origin, any vector in each of the three directions
being extended in a given ratio equal to the corresponding root;
or it may have two equal roots, in which case there corresponds
to the unequal root an invariant line, and to the multiple root
an invariant plane containing an invariant line, every vector
in the plane being multiplied by the root and then affected by
a shear of its points parallel to the invariant line in the plane;
or there may be three equal roots, in which case there is an
invariant line, a plane through this line, every line of the
plane through the origin being multiplied by the root and its
points sheared parallel to the invariant line, and finally every
line in space not in this plane is multiplied by the root and
its points sheared parallel to the invariant plane. In case
the function satisfies a reduced equation which is a quadratic,
this quadratic may have unequal roots, in which case there
is an invariant line corresponding to one root and an invariant
plane corresponding to the other, any line in the plane through
the origin being multiplied by the corresponding root; or there
may be two equal roots, in which case there is an invariant
plane such that every line in the plane is multiplied by the
root and every vector not in the plane is multiplied by the root
and its points displaced parallel to an invariant line. In case

22$ VECTOR CALCULUS

the reduced equation is of the first degree, every line is an
invariant line, all vectors being extended in a fixed ratio.
Where there are displacements, they are proportional to the
distance from the origin, and the region displaced is called a
shear region.

Hence <p takes the following forms in which gif g2, gz may-
be equal, or any two may be equal:

I. [g&SpyQ + g2pSya() + g^ySapOVSapy; reduced

equations for gx = g2 or gx = g2 = #3;
II. [9laSPyQ + giPSyaQ + mScfiO + a0Sj8y()]/So0y;

reduced equation for gi = g2, or if a — 0;
III. g + [(a/3 + cy)80yQ + bySya ()]/SaPy, reduced if
a = 0 = c, or a = 0 = b = c.

EXAMPLES
(1). Let <pp=V-app, where SaP + 0. Take X = a,
u = P, v = Vafi, then we find with little trouble

mi = - Sap, m2 = - a2/?2, ra3 = a2p2SaP,

and the characteristic equation of <p,

(tp + Sap)(<p - Tap)(<p + Tap) = 0.

Hence there are three invariant lines in general, and oper-
ating on p by (<p + Safi)(<p — TaP), we find the invariant
line corresponding,

(<p + SaP)p = aSpp + pSap,
(<p- TaP)(<p + Sap)p

= a2pSPp + P2aSap - aTaPSPp - pTaPSap
= - (TaSpp+ TpSap)(Ua+ UP)Tap.

Hence the invariant line corresponding to the root TaP is
Ua + Up. The other two are

Ua - Up and UVap.

THE LINEAR VECTOR FUNCTION 229

(2). Let <pp = Vafip.

(3). Let <pp = g2aS(3yp + frtfSyap + ySo&p) + hfiSaPp.

(4). Let <pp= gp+ (fifi + ly)Sfap + r(3Syap.

(5). Let <pp = Vep.

6. Solution of cpp = a. It is obvious that when <p satis-
fies the general equation

<p3 — mi<p2 + m2(p — ra3 = 0, ra3 4= 0,

then the vector

m%<p~lp = (w2 — miv? + <^2)p.

For if we take the <p function of this vector, we have an
identity for all values of p. Also this vector is unique, for
if a vector a had to be added to the left side, or could be
added to the left side, then it would have to satisfy the
equation <pa = 0. But if ra3 4= 0, there is no vector satis-
fying this equation, for this equation would lead to a
zero root for <p. Hence, if cpp = X, ra3p = m2X — mnp\ + <p2X,
which solves the equation.

If <p satisfies the general equation

(pi — mnp2 + m<2<p = 0, m% #= 0,

then we have one and only one zero root of the latent equa-
tion, and corresponding to it a unique vector for which
<pa = 0, and if (pp = X,

m2p = xa + [m\(p — (p2)p = xa + w&iX — <pX.

If (p satisfies the cubic

(pz — rriiv2 =0, mi 4 0,

the vacuity is two, and we have two cases according as
there is not a reduced equation, or a reduced equation exists

230 VECTOR CALCULUS

of the form <p2 — m\<p = 0. In either case the other root
is mi. There is a corresponding invariant line X, and if the
vector a is such that <pa = 0, then we have in the two cases
a vector (3 such that respectively <p(3 = a, or <p(3 = 0.
Hence, if <pp = 7, we must have in the two cases

7 = x\ + yot, or 7 = x\.

Otherwise the equation is impossible. Hence

mip = x\ + za + yj3 = 7 + ua -f 2/0,

where ^>/3 = a, <pa = 0, or where <pfi = 0 = ^ck-

If ^> satisfies the cubic

and no reduced equation, there are three vectors (of which
fi and 7 are not unique) such that <py = fi, <fP = a, <^a = 0,
and then <pp = X, we must have X = xa + yft where p is
any vector of the form

p = za + iCjS + 1/7.

If <p2 = 0, and no lower degree vanishes, then

<p(x(3 + 2/7) = <*j ^a = 0, and X = ua.

If <p = 0, there is no solution except for <pp = 0, where p
may be any vector.

7. Zero Roots. It is evident that if one root is zero,
then the region <p\ where X is any vector will give us the
other roots. For instance let <pp = Vep. Then if /x = Veh,

cpp, = Xe2 — eSe\, <p2fi = e2/x,

and the other two roots are ± V — 1 • Te.

If two roots are zero, then <p2 on any vector will give the
invariant region of the other root. For instance, let

THE LINEAR VECTOR FUNCTION 231

<pp = aSfiyp, then aSfiyaSfiyp = <p2p. Hence cpa = aSapy
gives the other root as Sapy and its invariant line a.

In case a root is not zero, but is g\, if it is of multiplicity
one, then <p — gi operating upon any vector will give the
region of the other root, or roots. If it is of multiplicity
two, then we use (<p — g{)2 on any vector.

8. Transverse. We define now a linear vector operator
related to <p, and sometimes equal to <p, which we shall
indicate by <p' ', and call the conjugate of <p, or transverse of
<p, and define by the equation

S\<pijl = Sn<p'\ for all X, /z.

For example, if <pp = Vap(3, then S\<pp = S\ap(3 = SpfiXa,
and <p' = VPQa = <p, if <pp = Vep, <p'p = — Vep; if
<pp = aSfip, <p'p = (3Sap.
If a is an invariant line of <p, (pa — ga, then for every /S

8p<pa = gSaP = Sa<p'P,
or

Satf - g)P = 0,

that is a is perpendicular to the region not annulled by
<p[ — g, that is invariant for <p' — g. If we consider that
from the definition we have equally

S\(p2p, = Sfxcp' X, S\(pziJL = Sup' X,

it is clear that <p and <p' have the same characteristic equa-
tion and the same general equation. They can differ only
in their invariant regions if at all. If then the roots are all
distinct, it is evident that the invariant line a of <p, is normal
to the two invariant lines of <p' corresponding to the other
two roots, hence each invariant line of <p is normal to the
two of <p' corresponding to the other roots, and conversely.
If now the characteristic equation is the general equation,

232 VECTOR CALCULUS

so that each function satisfies only the general equation,
let there be two equal roots, g, whose shear region gives

<pa= ga + ft <p@ = g(3, let <py = giy.

Then

&Vp = gSap + Sj3p, S/Vp = gSfip, Sycp'p = 0i#yp,
Sapy<p'p = g(V(3ySap + VyaSfo) + F)M0P

+ giVa(3Syp.

Therefore corresponding to the root g\, <p' has the in-
variant line Vafi, and to the root g, the invariant line V(3y.
Further (pf converts Vya into gVya + Vfiy.

Hence the invariant line of g\ for <p' is normal to the
shear region of g, and the shear region of g for <pf is normal
to the invariant line of g\ for <p, but the invariant line of
g for ip' is normal further to the shear direction of g for <p,
and the shear direction of <p' for g is normal to the invariant
line of (p for g.

In case there are three equal roots, and no reduced equa-
tion, we have

<pa = ga + ft <p/3 = gfi + 7, <£>7 = PY,

so that

&Vp = gSap + Sft>, Wp = gS(3p + S7p,

#7<p'p = gSyp,

Sapy<p'p = p^Sa/fy + VfhfSfo + F7CKS7P.

Hence, the invariant line of <p' is Vfiy, its first shear line
Vya, and second shear line Vafi.

In case there is a reduced equation with two distinct
roots, we have

<p(xa -f- y&) = 5f(ara + yfi), <P7 = 0i7,

Sa<p'p = gSap, S/Vp = gSfip, Sy<p'p = giSyp,

Sa&y-<p'p = gVfiySap + gVyaSfip + giVa(3Syp,

THE LINEAR VECTOR FUNCTION 233

Hence, the invariant line of <p' corresponding to gi is normal
to the invariant plane of g for <p, corresponding to g there
is an invariant plane normal to the invariant line of gi for <p.
Every line in the plane through the origin is invariant.
In case the reduced equation has two equal roots, then

<pa = ga + ft <pP = gP, <py = gy,

Sa<p'p = gSap + S(3p, Sy<p'p = gSyp, S(3<p'p = gSfip,

Sa(3y<p'p = gp + Sfa-iVfa),

Corresponding to g, we have then two invariant lines:
Va.fi, which is perpendicular to the shear plane of <p; V(3y,
which is perpendicular to the non-shear region of g and to
the shear direction of g; also the shear direction of <p' is
Vfiy, so that the shear region of <p' is determined by Vya
and Vfiy, and is therefore perpendicular to y.
The three forms of <p' are

I. <p' = [giVfySaO + toVyaSpQ + g3Va(3Sy01ISapy;

II. <p' = faVpySaQ + giVyaSpQ + aVfaSPQ

+ g2VaPSyQ]/Safrr,

III. <p' m g + [aVfiySPQ + bVyaSyQ + cVpySyQ]/SaPy.

We may summarize these results in the theorem :
The invariant regions of ip' corresponding to the distinct
roots are normal to the corresponding regions of the other
roots for <p. In case there are repeated roots, if there is a
plane every line of which through the origin is invariant,
then every line of the corresponding plane will also be in-
variant, but if there is a plane with an invariant line and
a shear direction in it, the first invariant line of the con-
jugate will be perpendicular to the shear direction and to
the second invariant line of <p, and the shear direction of the
conjugate will be perpendicular to the invariant lines of ip;

16

234 VECTOR CALCULUS

while finally, if there is an invariant line, a first shear direc-
tion, and a second shear direction, then the invariant line
of the conjugate mil be perpendicular to the invariant line
and the first shear direction of <p, the first shear direction
will be perpendicular to the invariant line and the second
shear direction of <p, and the second shear direction will be
perpendicular to the two shear directions of <p. Let a, /3, y
define the various directions a = V(3y/Sa(3y, /? = Vya/Sa/3y,
y = VaP/Sa(3y, then we have

<p = gioSoi + gtffS0 + gzySy)
<p' = giaSa + g2@S(3 +. g3ySy J

or

or

(giaSa + gJISp + afiSa + g2ySy)
\giaSa + gi(3S(3 + aaS(3 + g2ySy\

ig+(aJl+cy)Sa+byS(3
\g+aaSp+ (b(3 + ca)Sy .

9. Self Transverse. It is evident now that <p = <p' only
when there are no shear regions, if we limit ourselves to
real vectors, and further the invariant lines must be per-
pendicular or if two are not perpendicular, then every
vector in their plane must be an invariant, and even in this
case the invariants may be taken perpendicular. Hence
every real self-transverse linear vector operator may be
reduced to the form

<pp = — aSapgi — (3S(3pg2 — ySypg3,

where a /3 y form a trirectangular system, and where the
roots g may be equal.

Conversely, when <p — <p', the roots are real, provided
that we have only real vectors in the system, for if a root
has the form g + ih, where i — V — 1, then if the invariant

THE LINEAR VECTOR FUNCTION 235

line for this root be X + ip,, where X and p are real, we have
<p(\ + in) = (g + ih)(k + ifi) = g\ — hp + i(h\ + gp)

= <p\-\- iipjJL.

Therefore

<p\ = g\ — hp, <pp = hX + gfx,
and

$/*<pX = gS\p — hp2 = S\<pp = AX2 + <7$Xju.

Thus we must have

^X2 + hf? = 0.

It follows that h = 0.

Of course the roots may be real without <p being self-
transverse.

An important theorem is that <p tp' and <p'<p are self-
transverse. For

Sp<p(p'(T = Sa<p<p'p, Sp<p'<p<r — S(T(pf(pp.

EXERCISE

Find expressions for <p<p' and <p'<p in terms of a, /3, 7, a, jS, 7.

10. Chi of p. We define now two very important func-
tions related to <p and always derivable from it. First

X* = mi — <P>
so that

Sa(3y-x<pP — pSafi<py + pS(3y<pa + pSyapfi — (paSfiyp

— cpfiSyap — <pySa(3p
= VVaPV<pyp + • • •
= aSp((3<py — y<p(3) + • • •.

The other function is indicated by \j/^ or by xvv and defined

4rp = m2 — mi(p + <p2 = ra2 — <pxv,
Sapy-i/z^p = pSonpfiipy + • • • — <paSp((3(py — y<p(3)
= aSp<p(3(py -\~ f3Sp<py<pa + ySp<pa<p(3.

236 VECTOR CALCULUS

We have at once from these formulae the following im-
portant forms for FX/x,

X„FX/x = [aSVlniVpvy - Vy<pp) • • .]/SaPy

= [aS(V<p'\n - V\<p'n)V0y + • • -]/SaPy
= *W + V\i/>%

Whence we have also

<pV\p = miV\ix — V\<p'n — Vp'XfjL,
1^FX/x = [aSV\»V<pp<py H ]/Sapy

= V<p'\<p'lL.

Since it s evident that

X+ = x/j and \pv, = #/,
we have at once

x\V\ii = V<p\» + V\<pfi

^FX/x = V(p\<pji.

The two expressions on the right are thus shown to be
functions of FX/x.

It is evident that as multipliers of p

™<i = <P + X = f' + X'i
^2 - *>X + lA = *>'x' + ^',

m3 = ^ = ^V-

EXERCISES

1. If <p = aiSPiQ + a2SM) + «*Sfo(), show that

<p' = faScHQ + 02Sa2() +01&I.O,
X = 27/9i7ai(),
* = - 2F/31/825Fala2(),
mi = 2*Sau9i, ra2 = — Z£Faia2F(8i/32, ra3 = — Saia2azS0ifi203,

X' = UFaiV/SiO,
^ = - HVaiatSVPiPiQ.

2. Show that the irrotational invariants of x and ^ are mi(x) = 2mh
m2(x) = »ii2 + m2, w3(x) = Wim2 — m3; rai(^) = ra2, m2(^) = raira3,
Wj(^) = m32.

THE LINEAR VECTOR FUNCTION 237

3. For any linear vector function <p, and its powers <p2, <p3, • • • , we have

mi(<f?) = Wi2 — 2ra2, m2(<p2) = m22 — 2raira3, w3(^) = ra32.
mi(<p3) = mi3 — 3wiW2 + 3m3, m2(<p3) = 3raira2ra3 — m23 — 3ra32,

m3(<p3) = m33.
mi(^4) = mi3 — 4mi2w2 + 2w22 + 4raim3
w2(<p4) = w24 — 4wiw22w3 + 2wi2m32 + 4ra2ra32, m3(^>4) = ra34.

4. Show that for the function <? + c, where c is a scalar multiplier,

mi(<p + c) = wi(^) + 3c, m2((p + c) = 0ts(?) + 2mi(<p)c -f 3c2,
wi3(¥> + c) = ws(«p) + cw2(<p) + c2mi(<p) + c3.

5. Study functions of the form x\p + ?/x + 2.

6. <p'V<p\<pfi = m3V\n; <p'(V\<pn — F/x^X) = m2F\M — V<p\<pn.

7. ^(a^>) = aHiv)', tifilPi) = ^(<Pi)-^('Pi)'

8. «A(a) = a2, ^[7a()J = - aSaQ, *(- 0Sa) = 0.

+{— QxiSi - gsjSj - g3kSk) = - g2g3iSi — g3gijSj - gig2kSk.

= - VptfiSVaiyi - VPtppSVata, - VptfiSVa&n.

10. For any two operators <p, 9,

mi(<pd) = mi(M, m.2(<pd) = m2(6<p), m3(<pd) = tr»»(0*).
mi((p6) = mi(<p)mi(0) + ra2(<p) + ra2(0) — m2{6 + <p).
m2(<pd) = m2{6)m2(<p) + m3{<p)-mv{d) + ra3(0)-rai(y?)

- mtffto + *'(*)].
m3(<pd) = m3(<p)-m3{6).
rrii(<p + 0) = mi{<p) + Wi(0).

m2(v? + 0) = m2(v?) + m2(0) + mi(6)'ini(ip) - nii(<pO).
mt(<p + 0) = m3(*>) + m3(0) + mi[*V(4) + 0V(*>)].

11. x can have the three forms :

t (ff* + 9t)*Sa + (g3 + 0i)0S0 + (oi + g2)ySj;
II. fo + o2)a£5 + fo + fln)/aSg + 2^x7^7 + apSa;
III. 2g - (a/3 + c7)>S£ - bySfT

The operator x is the rotor dyadic of Jaumann.

12. The forms of \f/ for the three types are

I. g2ggaSa + g3gi&Sl3 + gig2ySy;
II. gig2aSZ + g2gi0S8 + 0i27#t" - agtfSZ;
III. o2 - [O03 + (ab - gc)y]Sa - bgySfi.

238 VECTOR CALCULUS

13. An operator called the deviator is defined by Schouten,* and is
for the three forms as follows:

I. (l9i - 9* - gs)aSZ + (Itfi - 0* - gi)0S8 + (lg* - gx - g%)ySy';
II. (- fci - <7*)(«S5 + fiSfi) + (§0» - 2^)7^ + apSZ;
III. (o£ + Cy)Sa + bySd.

It is V<p = <p — S<p, where S(<p) = \m\.

14. Show that if F (X, M) - - F (m, X) then

F (X, M) - C (X, m)Q.VXm,
where C is symmetric in X, n and Q is a quaternion function of VX/i.
11. We derive from <p and ^' the two functions

That there is a vector e satisfying this last equation, and
which is invariant, is easily shown. For if we form

™>z(<P — <p')> we find that

S(<p - <p')\{<p - <p')n(<p - (p')v

= S(p\<piJL<pi> — 2$ (p\<p' )jl<p' v — ^LSipkcpyup'v—Sip'Xv'mp'v
= S\iAi>(m3 — ra3 + Wi(^» — mi(^')).

But it is easy to see that this expression vanishes identically,
for the first two terms cancel, and if <plt <p2 are any two linear
vector functions, we have

= Siiv<p\kSiiv<p<Lh + SjjLvcpiiiSvXip^K + SiiV(pivS\mp2K

+ Sl>\<Pi\SlJlV<p2lJL + Sv\<PillSv\(p2lJL + Sp\<PipS\/JL(P2H
+ S\fJL<Pi\SlJLV<p2V + S\lJL<PilJ,ST<P2V + $XjU<pi J/jSAjU^

= S^knv - mi(<p2<pi) .

Hence we may under mi permute cyclically the vector
functions. Again after this has been done we may take
the conjugate. Hence the expression above vanishes, and
there is a zero root in all cases for <p — <p'. Further we
may always write

* Grundlagen der Vector- und Affinor-Analysis, p. G4.

THE LINEAR VECTOR FUNCTION 239

S\fXV<pp = (pXSfJLPp + • • *

S\jJLl>'<p'p = VjivS\<pfp + • • •
= VfivSipKp +
Hence we have

S\nv(<p - <p')P = VPV(Vfip)cp\ + ....

From this we have 2eS\pv = V(p\Vpv + • • • for every
noncoplanar X, p, v.

The function <p0 is evidently self-transverse, and the
conjugate of VeQ is — VeQ. It is easy to show that

2<peS\pv = — V\V<pp(pv — • • •.

The expressions Te, T<pe, and Sepe are scalar invariants
of <p, and these three may be called the rotational invariants.
In terms of them and the other three scalar invariants all
scalar invariants of <p or <p' may be expressed.

If there are three distinct roots, g\, g2, g3, and the corre-
sponding invariant unit vectors are yh y2, 73, we may set
these for X, p, v, and thus

2e&7iy2Y3 = giVjiVy2y3 + g2Vy2Vyzyi + gzVyzVy^y2
= (92 — g3)yiSy2y3 + (g3 - gi)y2Sysy2

+ (gi — g2)yzSyiy2.
2<peSyiy2y3 = - g2g3VyxVy2yz — g3g\Vy2Vyzyx

— gig2Vy3Vyiy2.

In case two roots are equal and (pa = gxa + (3h2,
<p(3 = #i/3, (py = g2y, we have

2eSa(3y = (g2 - gi)VyVafi + VQVPyh.

In case three roots are equal, <pa = ga-\-h(3, <pfi = gr/5+ ly,
<py = gy

2eSa(3y = h(3V(3y + lyVya.

It is evident, therefore, that if the roots are distinct and

240 VECTOR CALCULUS

the axes perpendicular two and two, that « = 0; if two
roots are equal and the invariant line of the other root is
perpendicular to the plane of the equal roots, then it is the
direction of e; and if the three roots are equal, and if the
invariant line is perpendicular to the two shear directions,
then € is in the plane of the invariant line and the second
shear.

12. Vanishing Invariants. The vanishing of the scalar
invariants of (p leads to some interesting theorems.

If Wi = 0, there is an infinite set of trihedrals which are
transformed by <p into trihedrals whose edges are in the
faces of the original trihedral. If ^transforms any trihedral
in this manner, mi = 0, and there is an infinite set of trihe-
drals so transformed.

We choose X, n, v for the edges of the vertices, and if <p\
is coplanar with /z, 7, <pix with v, X, and <pv with X, ju, the
invariant mi = 0. If mi = 0, we choose X, ju, arbitrarily,
and determine v from ScpXnv = 0 = SXcpuv. Then also
S\n<pp = 0.

The invariant m2 vanishes if <p transforms a trihedral
into another whose faces pass through the edges of the first.
The converse holds for any infinity of trihedrals.

EXERCISES

1. Show that if a, fi, 7 form a trirectangular system

mi = — Sa<pa — S/3<pl3 — Sy<py

and is invariant for all trirectangular systems,

m2(<p<p') = T*<poc + TV/? + T*<py,
TV(X) = S2\<pa -f £2Xv/3 + S2\<py.

2. Study the functions for the ellipsoid and the two hyperboloids

- <p = a^aSa ± b~2fiSl3 ± c^ySy.
3 Study the functions

ZmVaVQct, <P + VaVQa, a^VoapQ,

r-VpVaQ, V -vVaQ.fi.

THE LINEAR VECTOR FUNCTION

241

4. Show that

V <pp = 2e — mi,
\/Sp<pp = — 2 (pop,

VAp = — 2<pt — m2,
\7Vp<pp = 2Sep + Wip — 3<pp,

wherein <p is a constant function. Hence (pop may always be repre-
sented as a gradient of a scalar, Sep as a convergence of a vector, and
m,\p — 3<pp (deviation) as a curl. We may consider also that Wi is a
convergence and e is a curl, ra2 a convergence and <pe a curl.

5. An orthogonal function is defined to be one such that

ip<p' = 1.

Show that an orthogonal function can be reduced to the form

ip = () cos 0 - sin 0-70/3 = (lT cos 9)0800 = 0**l*Q0-*l*

or — /3(0/"-)+1()/?-(fl./7r)-1 which is a rotation about the axis /3 through
the angle — 0, or such a rotation followed by reflection in the plane
normal to /?.

6. Study the operator <p112.

7. Show that

m

■i(<po) = mh

m2

(<Po) =

mi + e2, m3(<p0) = m3 ■

f 5-6V56.

Hence if
Tf

Te

= 0,

m2(<p) = m2(<p0).

u

8.

Show that

Se<pt

i = 0,

m3(<p) = m3(<p0).

mAVe{)]

= o,

m*[VeQ] = TV, m3[Ve()] =

0.

9.

Show that

e(x) = -

■ 6,

«(x)

- " *e' e(^_1) = " m3

ipe.

10,
11,

, Show that

. if 0 = y./M),

\p(<Po)

= *0 + aSe().

rni(d) =

2S/3e,

m-i

(0) = - S/3*>/3, m3(d) =

o,

"2(a2 — aSa),

12. If *> = F-a(),

,p2» =

^2n+l = a2n7a()>

13. For any two operators <p, 0,

2eM) = 2e(^o0o) + X(<p)e(6) + x{0)e(<p) + V-e(<p)e(6).

242 VECTOR CALCULUS

In particular

14. An operator ^> is a similitude when for every unit vector a,
T^a = c, a constant.

Show that the necessary and sufficient condition is

<p'<p = c2.

Any linear transformation which preserves all angles is a similitude.

15. If <p = aSi + 0Sj + ySk, then <p' = iSa +jSp + kSy, and
^j^' = — ctSa — fiSfi — 7$7,

mi(*V) = Pa + P/3 + 7*7, m,(^^') = PFa/S + 7*7/37 + T^a,
mz(<p<p') = — S2a0y.

13. Derivative Dyadic. There is a dyadic related to a
variable vector field of great importance which we will
study next. It is called the derivative dyadic, since it is
somewhat of the nature of a derivative, as well as of the
nature of a dyadic. This linear vector function for the
field of a will be indicated by Da and defined by the equation

D.= ~ SQV-<r.

It is evident at once that if we operate upon dp, we arrive
at da. This function is, therefore, the operator which en-
ables us to convert the various infinitesimal displacements
in the field into the corresponding infinitesimal changes
in the field itself.
The expression

SdpDJp = Cdf,

where C is a constant and dt a constant differential, repre-
sents an infinitesimal quadric surface, the normals at the
ends of the infinitesimal vectors dp being Dadp.

Let us consider now the field of a, containing the con-
gruence of vector lines of <r. Consider a small volume
given by 8p at the point whose vector is p, and let us sup-

THE LINEAR VECTOR FUNCTION 243

pose it has been moved to a neighboring position given by
the vector lines of the congruence, that is, p becomes
p + adt. Then p + 8p becomes

p+8p + dt(<r + DM,

that is to say, dp has become

(1 + Dadt)8p.

Hence any area V8\p82p becomes, to terms of the first
order only,

V8lP82p + dt(V8lPD,82p + VDJlP82P).

The rate of change with regard to t of the vector area
V8ip82p is therefore

X (Dff)V8lP82p.

Likewise, the infinitesimal volume S8ip82p8sp is trans-
formed into the volume

S8ip82p83p + dt{S8ip82pDa8zp + S8ipDa82p83p

+ SDa8ip82p83p).

The rate of increase of the volume is, therefore, miS8ip82p83p.
In other words if we displace any portion of the space of
the medium so that its points travel infinitesimal distances
along the lines of the congruence of a, by amounts propor-
tional to the intensity of the field at the various points, then
the change in any infinitesimal line in the portion of space
moved is given by dtDff8p, the change in any infinitesimal
area is given by x'(Da)dt- Area, and the change in an
infinitesimal volume is midt times the volume.

In case a defines a velocity field the changes mentioned
will actually take place. We have here evidently a most
important operator for the study of hydrodynamics. If
adt is the field of an infinitesimal strain, then Da8p is the

244 VECTOR CALCULUS

displacement of the point at dp. Evidently the operator
plays an important part in the theory of strain, and con-
sequently of stress. Further, (we shall not stop to prove
the result as we do not develop it) for any vector a a
function of p we have an expansion analogous to Taylor's
theorem, in the series

h2
<r(p0 + ha) = (r(po) + hD^ + -^ (- &*V)Z).a

+ | (SaV)2Daa + ••••

This formula is the basis of the study of the singularities
of the congruence. For if cr(p0) = 0, then the formula will
start with the second term, and the character of the con-
gruence will depend upon the roots of Dff. In brief the
results of the investigation of Poincare referred to above
(p. 38) show that if none of the roots is zero, we have the
cases :

1. Roots real and same sign, the singularity is a node.

2. Roots real but not all of the same sign, a faux.

3. One real root of same sign as real part of other two,
a focus.

4. One real root of sign opposite the real part of others,
a faux-focus.

5. One real root, other two pure imaginaries, a center.
If one or more roots vanish, we have special cases to con-
sider.

The invariants of Da are easily found, and are

mi = — SV<r, e = ^Vxja, m2 = — %SVViV2V<tkt2,
D*e = iV-VViV2V<ri<T2, m3 = |SViV2V3<W-20-3.

After differentiation, the subscripts are all removed. The
related functions are

THE LINEAR VECTOR FUNCTION 245

BJ = - v&r(), X = - VVV*Q, %' = ~ V-VQV-*,

$ = - jrviV2^K72(), y = - i&oviVs-Wi*!.

In a strain a the dilatation 's rab the density of rotation
(spin) is e, and in other cases we can interpret m\ and e in
terms of the convergence and the curl of the field. In
case a is a field of magnetic induction due to extraneous
causes, and a is the unit normal of an infinitesimal circuit
of electricity, then %'« is the negative of the force density
per unit current on the circuit. In any case we might call
— x'V8iP82p the force density per unit circuit. Since x'
is not usually self-transverse, the force on circuit a has a
component in the direction jS different from the component
in the direction a of the force on circuit ft.

Recurring to Stokes' and Green's theorems we see that

fdpa = ff - WVdlPd2P'(T

= 2ffS8lP82pe -Sfx'V8lP82P.

It is clear that the circulation in the field of a is always
zero unless for some points inside the circuit e is not zero.
The torque of the field on the circuit vanishes for any
normal which is a zero axis of x'« From these it is clear that,
if we have a linear function <pdP, in order that it be an exact
differential da we must have the necessary and sufficient
conditions

VVvO m 0.

For if tf<pdP = 0, then <pVUvV = 0 for all Uv, whence
the condition. The converse is easy.

The invariant m3 in the case of the points at which a = 0
will be sometimes positive, sometimes negative. A theorem
given originally by Kronecker enables us to find what the
excess of the number of roots at which ra3 is positive over
the number of roots at which ra3 is negative is.* We set

* Picard, Traite d'Analyse, Vol. I, p. 139.

246 VECTOR CALCULUS

t = fa/To* and 7 = - J- ffSdvr]

47T

then the integral will vanish for any space containing no
roots, and will be the excess in question for any other space.
We could sometimes use this theorem to determine the
number of singularities in a region of space and something
about their character. It is evident that <SVr = 0.

The operator (Dc)0 = \(D0 + DJ) is called the deforma-
tion of the field, and the operator Ve() the rotation of the
field.

In case a is a unit vector everywhere, then DJa = 0,
and since the transverse has a zero root, Da itself must have
a zero root. There is one direction then for which Dffa = 0.
The vector lines given by Vadp = 0 are the isogons of the
field. In case there are two zero roots the isogons are any
lines on certain isogon surfaces.

EXERCISES

1. Study the fields given by

a = — p, a = Up/p2, a = Vap, a = aSfip, a = Vap/p3.

2. Show that if a is a function of p,

a + da = — V o[Spo<r — %Spo<ppo] — \V Po^V <*
= VVoihVvpo - Wpo<PPo] - lSV<r,

where Vo operates only on p0, and <p = — <rSS7 0- The first form
expresses a + da as a gradient and a term dependent on the curl of a,
the second as a curl and a term dependent on the convergence of a.
po is an infinitesimal vector.

3. If a = FVr, Da = ZV.

14. Dyadic Field. If <p is a linear vector operator de-
pendent upon p, we say that <p defines a dyadic field. For
every point in space there will be a value of <p. Since there
is always one root at least for <p which is real, with an in-
variant line, there will be for every point in space a direction

THE LINEAR VECTOR FUNCTION 247

and a numerical value of the root which gives the real
invariant direction and root. These will define a con-
gruence of lines and a numerical value along the lines.
In case the other axes are also real, and the roots are distinct
or practically distinct, there will be two other related con-
gruences. The study of the structure of a dyadic field
from this point of view will not be entered into here, but
it is evidently of considerable importance.

EXERCISES

1. If <p = uQ, then the gradient of the field is Vw. The vorticity
of the field is VV <p() — VVuQ. The gradient in any case is v'V,
a vector.

2. If <p = VaQ, the gradient is — V\7<r, the vorticity is

QSS7*+DV m - x{D,r).

3. If <p = <tSt(), the gradient is aSVr — DT<r, the vorticity is
WvStQ + V<rDy(). The gradient of the transverse field is tS\7<t

- DaT, the vorticity VX/tSvQ + VtD<j{).

4. If ip = VadQ, the gradient is - 70(V)<r + VadV, the vor-
ticity is

S\7<T-d() +S*V'-0'Q -*S$[(V i)Q +£<r0().

For the transverse field we have

the gradient is — 0'FV'o- — 0VV<r,
the vorticity 7v W«r() + W'dVa'Q.

5. If <p = D,r the gradient of the field is — VV, the concentration of
<r, and the vorticity is D vva • The gradient of the transverse field is

— V^Vo", while the vorticity is zero.

6. If <p = VV0(), the gradient is FV0V, where both V's act on 0,
and the vorticity is V20() - V&V0().

7. If <p = De(a), the gradient is — 7^, the vorticity is Dvve<r-

8. If f> = to, the gradient is 2e(07V0).

9. For any <p

Vm1 = <PV +2e (FWO ),
Vm2 = 2 € {<pW<p' + FW' 0 I,
Vw3 = 2 e [V (Vi M *'
-t W Vx].

248 VECTOR CALCULUS

15. The Differentiator. We define the operator — SQ V
as the differentiator, and indicate it by D. It may be used
upon quaternions, vectors, scalars, or dyadics.

As examples we have, D being the transverse

Bv„ = VaDr() - VtD0Q, DSar - SQD.r + S()DTa,
DVaa = - VaD.Q, DmiM = mriDJ,

DeM= e(DJ, Dv= -S()V •*>().

16. Change of Variable. Let F be a function of p, and
p a function of three parameters u, v, w. Let

A = ad/du + f3d/dv + yd/dw,

where a, /3, y form a right-handed system of unit vectors.
Then we have the following formulae to pass from expres-
sions in terms of p to differential expressions in terms of the
parameters.

AF = - AiSPlVFt
FA' A" = |FAi'A2"£Fpip2Fv'V",
SA'A"A'" - - i<SAi'A2"A8,"iSpiP2PsSV' V'V".

As instances

- SVv= A'VV'V,
VA<r= VV"T(r"A'.

Notations
Dyadic products
4>(a), <f>'(a), <f)Va( ), Va(f>( ), Hamilton, Tait, Joly, Shaw.
<l>'af a-4>, <j) X a, aX <j>, Gibbs, Wilson, Jaumann, Jung.

Reciprocal dyadic
4>~l, Hamilton, Tait, Joly, Gibbs, Wilson, Burali-Forti,

Marcolongo, Shaw.
q~l, Timerding.
I6I"1, filie.

THE LINEAR VECTOR FUNCTION 249

The adjunct dyadic
\j/ = m(f)'~l, Hamilton, Tait, Joly, Shaw.
WO2, Gibbs, Wilson, Macfarlane.
R{a), Burali-Forti, Marcolongo.
x((f>, (f>), Shaw.
D4>~1, Jaumann, Jung.

The transverse or conjugate dyadic
<f>', Hamilton, Tait, Joly.
0, Taber, Shaw.

<f>c, Gibbs, Wilson, Jaumann, Jung, Macfarlane.
K(ct), Burali-Forti, Marcolongo.
\b / , Elie.

The planar dyadic
X = Wi — (f>r, Hamilton, Tait, Joly.
4>J — <f>c, Gibbs, Wilson.
— </>/, Jaumann, Jung.
CK(a), Burali-Forti, Marcolongo.
x(0), Shaw.

Self-transverse or symmetric part of dyadic
<f>ot Hamilton, Tait, Shaw.
$, Joly.

<f>f, Gibbs, Wilson.
[</>], Jaumann, Jung.
D(a), Burali-Forti, Marcolongo.
\ b /, Elie.
\ b° / , Elie. In this case expressed in terms of the axes.

Skew part of dyadic
\{4> — </>') = V-e( ), Hamilton, Tait, Joly, Shaw.
</>", Gibbs, Wilson.
II, Jaumann, Jung.
Va A , Burali-Forti, Marcolongo.

17 i

250 VECTOR CALCULUS

\ b / , £lie.

Sin <f>, Macfarlane.

Mixed functions of dyadic
X«>, 0), Shaw.
\<f>l 0, Gibbs, Wilson.
R{(f>, 0), Burali-Forti, Marcolongo.

Vector of dyadic
e, Hamilton, Tait, Joly.
<£x, Gibbs, Wilson.
(f>r8, — <}>/, Jaumann, Jung.
Va, Burali-Forti, Marcolongo.
E, Carvallo.
R = Te, filie.
c(<£), Shaw.

Negative vector of adjunct dyadic
<f>e, Hamilton, Tait, Joly.
0-0x, Gibbs, Wilson.
<t>-<f>r8, Jaumann, Jung.
olVol, Burali-Forti, Marcolongo.
«x(</>> <f>)> Shaw.

Square of pure strain factor of dyadic
4><f>', Hamilton, Tait, Joly.
</></>c, Gibbs, Wilson.
{(f)}2, Jaumann, Jung.
aKa, Burali-Forti, Marcolongo.
[6], filie.
</></>', Shaw.

Dyadic function of negative vector of adjunct
<f>2e, Hamilton, Tait, Joly, Shaw.
<f>2-4>x, Wilson, Gibbs.

THE LINEAR VECTOR FUNCTION 251

02 -0/, Jaumann, Jung.

a2Va, Burali-Forti, Marcolongo.

K2, Elie.

Scalar invariants of dyadic. Coefficients of characteristic

equation
m" ', ra', m, Hamilton, Tait, Joly, Carvallo.
1%, h, h, Burali-Forti, Marcolongo, Elie.
F, G, H, Timerding.
0S, (</>2)s, 03, Gibbs, Wilson,
mi, ra2, ra3, Shaw.

fc, ]

4>8*, >■ • • • 03, Jaumann, Jung.

- w, J

cos </>••• 03, Macfarlane.

(Mer scalar invariants
™>i(<f>o2), mi(00'), 2(rai2 — m2), rai(00')>

wi[x(0, *)> 0L Shaw.
[08]2«, {0js2> [01/, •'* -j Jaumann, Jung.
• • •, • • •, • • ., 0 : 0, 0* 0 : ft Gibbs, Wilson.
Elie uses ifi for $e0e.

Notations for Derivatives of Dyadic
In these V operates on 0 unless the subscript n indicates
otherwise.

Gradient of dyadic
V0, Tait, Joly, Shaw.

Dyadic of gradient. Specific force of field
0V, Tait, Joly, Shaw,
grad a, Burali-Forti, Marcolongo.

-3 — , Fischer.
dr

252 VECTOR CALCULUS

Transverse dyadic of gradient
0'V, Tait, Joly.
grad Ka, Burali-Forti, Marcolongo.

—r^-y Fischer.

V -<t>, Jaumann, Jung.

Divergence of dyadic

- SV<f>( ), Tait, Joly, Shaw.

X grad Ka, Burali-Forti, Marcolongo.

Vortex of dyadic
VV4>( ), Tait, Joly, Shaw.
Rot a, Burali-Forti.

V X 0, Jaumann, Jung.

Directional derivatives of dyadic

- S( ) V • 0. Sa'1 V ■ <l>a. ScT1 V -<t>Va(), Tait, Joly, Shaw.
S(a, ( )), Burali-Forti.

P , IX*, Fischer.
da da

Burali-Forti, Marcolongo.

(»<>)<»•

Gradient of bilinear function
ju„(Vn, «), Tait, Joly, Shaw.
<£(/z)a, Burali-Forti.

Bilinear gradient function
ju(Vn, un), Tait, Joly, Shaw.
\//(n, u), Burali-Forti.

Planar derivative of dyadic
<f>nVVn( ), Tait, Joly, Shaw.

X-^> Fischer,

CHAPTER X
DEFORMABLE BODIES

Strain

1. When a body has its points displaced so that if the
vector to a point P is p, we must express the vector to the
new position of P, say P', by some function of p, cpp,
then we say that the body has been strained. We do not
at first need to consider the path of transition of P to P'.
If cp is a linear vector function, then we say that the strain
is a linear homogeneous strain. We have to put a few
restrictions upon the generality of <p, since not every linear
vector function can represent a strain. In the first place
we notice that solid angles must not be turned into their
symmetric angles, so that SipKcpyupvlSKp.v must be positive,
that is, ra3 is positive. Hence (p must have either one or
three positive real roots. The corresponding invariant lines
are, therefore, not reversed in direction.

2. When <p is self-conjugate there are three real roots
and three directions which form a trirectangular system.
The strain in this case is called a pure strain. Any linear
vector function can be written in the form

#rr V.{**9."fI0« = p-1V(<pV)()-p,

where

q-i()q = (wT'V

The function <p<p' is self -conjugate and, therefore, has three
real roots and its invariant lines perpendicular. If we set
7r = V ((p<p')y then 7r2 = ipip'. Let the cubic in <p<p' be
G3 - MXG2 + M2G - M3 = 0. Then from the values
given in Chapter IX, p. 237, for the coefficients of <p2

253

254 VECTOR CALCULUS

in terms of those of <p we have (the coefficients of the cubic
in w being pu p2, p3)

Mi = pi* - 2p2f M2 = p22 — 2pm, M3 = pz2,
whence we have

P!4 - 2(Mi + 8M3)pi2 - \m2MzVl + MS - 4M22M3 = 0.

Thence we have pi, p2, and p3.

Now if the invariant lines of <p<p' are the trirectangular
unit vectors a, 0, 7, we may collect the terms of <p in the form

<P = aaSa'Q + bpSP'Q + cySy'Q,

where a, b, c are the roots of V <p<p' = w and a! , fi', 7' are
to be determined. Hence <p' = aa'SaQ + • • • and

- tp'ip = tfa'Sa'Q + VP'Sp'Q + WO-

But also

^' = _ otefo _ fc0S0 - c27#7,

since a, /?, 7 are axes of <p(p', and a2, b2, c2 are roots. Now
we have

<p'a = — act', <p'/3 = — b(3', <p'y = — cy',

hence

<p(pra = a2a = — a2otSa'ct' — ab(3Sa'(3' — acySot'y'.

Thus we have a:'2 = — 1, Set' (3' = 0 = Sa'y', and similar
equations, so that a', (3', y' are unit vectors forming a tri-
rectangular system, and indeed are the invariant lines of
<p'<p. We may now write at once

7r = — aaSct — b/3S(3 — cySy,
q~\)q = - aSa' - fiSfi* - ySy'.

This operator obviously rotates the system a', (3', y' into

DEFORM ABLE BODIES 255

the system a, (3, y, as a rigid body. That the function is
orthogonal is obvious at a glance, since if we multiply it
by its conjugate we have for the product

- aSa - PSP - ySy = 1().

Reducing it to the standard form of example five, Chapter
IX, p. 236, we find that the axis is UV(aa' + (3(3' + 77') and
the sine of the angle of rotation \TV{aa! + $8' + 77') •

EXAMPLES

(1). Let <p = VeQ- Then

<p' = - VeQ, <P<P' = ~ VeVeQ = eSe() - e2.

The axes are e for the root 0, and any two vectors a, j8
perpendicular to e, and these must be taken so that a(3 = Ue,
the roots that are equal being T2e. We may therefore write

<p = TeaS(3 - Te(3Sa = V-eQ,

which was obvious anyhow. Hence we have for q~lQq the
operator

aSp-0Sa= V(VaP)Q,

and this is a rotation of 90° about Va(3 = Ve of 90°. The
effect of

7T = Te(- aSa - (3S(3)

is to give the projection of the rotated vector on the plane
perpendicular to e, times Te. That is, finally, VeQ rotates
p about € as an axis through 90° and annuls the component
of the new vector which is parallel to e.

(2). Consider the operator g — aS(3Q where a, jS are any
vectors. It is to be noticed that we must select of all the
square roots of (p<pf that one which has its roots all positive.
It is obvious that j) = q.

256 VECTOR CALCULUS

3. The strain converts the sphere Tp= r into the ellip-
soid 7V-1p = r, or

WW = - r\

This is called the strain ellipsoid. Its axes are in the direc-
tions of the perpendicular system of (p<p' — tt2. The ellip-
soid Sp<p'<pp = — r2 is converted into the sphere Tp = r.
This is the reciprocal strain ellipsoid. Its axes are in the
directions of the principal axes of the strain. The exten-
sions of lines drawn in these directions in the state before
the strain are stationary, and one of them is thus the maxi-
mum, one the minimum extension.

4. A shear is represented by

<PP — P ~ fiSap,

where Sa/3 = 0. The displacement is parallel to the vector
/3 and proportional to its distance from the plane Sap = 0.
There is no change in volume since ms = 1.

If there is a uniform dilatation and a shear the function is

<pp = gp- fiSap.

The change in volume is now g3. The equation is easily
seen to be

(<P - 9? = 0.

This is the necessary and sufficient condition of a dilatation
and a shear, but this equation alone will not give the axes
and the shear plane, of course.

5. The function <pp = gqpcT1 ~ qfiq~lSap is a form into
which the most general strain can be put which is due to
shifting in a fixed direction, U(5, planes parallel to the fixed
plane Sap = 0 by an amount proportional to the perpen-
dicular distance from the fixed plane, then altering all
lines in the ratio g} and superposing a rotation. This is

DEFORMABLE BODIES 257

any strain. We simply have to put <p'<p into the form

<p'<p = b2 + X£ju + ftSk,
where

S\fx = i(a2 + c2- 2b2), T\n = K«2 - c2),
and then we take

g=b, a - - X, bp = n- IXrK* ~ c)2-

The rotation is determined as before.

6. All the lines in the original body that are lengthened
in the same ratio, say g, are parallel to the edges of the cone
TcpUp = g or SUp(<pf<p — g2)Up = 0, or in terms of X, /z,
2SX UpSfx Up = b2 - g2, sin u-smv= (b2- g2)/(a2 - c2) ,
where u and v are the angles the line makes with the cyclic
planes of the cone Scpp<pp = — T2p.

7. The displacement of the extremity of p is

5 = a — p = O — l)p,

which can be resolved along p and perpendicular to p into
the parts

p(/Sp~Vp — 1) + pVp~l<pp.

The coefficient of p in the first term is called the elongation.
It is numerically the reciprocal of the square of the radius
of the elongation quadric:

Sp(<p0 — l)p = — 1,

the radius being parallel to p.

The other component may be written Vep + Vcpopp~l- p,
where e is the invariant vector of <p, the spin-vector.

8. If now the strain is not homogeneous, we must con-
sider it in its infinitesimal character. In this case we have
again the formula da = — SdpV -tr = cpdp, where a is now
the displacement of P, whose vector is p, and a + da that of

258 VECTOR CALCULUS

p -f- dp, provided that we can neglect terms of the second
order. If these have to be considered,

da = - SdpV a + i(SdpV) VV
= (pdp — %SdpV - (pdp.

We may analyze the strain in the case of first order into

<P = (fo + VeQ.

Since now € = \V\7<r, if e = 0, it follows that a = VP
and there is a displacement potential and

p '« - VSVP().

The strain is in this case a pure strain. If e is not zero,
there is rotation, about e as an axis, of amount Te. In any
case the function <p0 determines the changes of length of all
lines in the body, the extension e of the short line in the
direction Up being

— SUppoUp.

The six coefficients of <p0, of form — Sa<po(3, where a, ft
are any two of the three trirectangular vectors a, ft 7,
are called the components of strain. Three are extensions
and three are shears, an unsymmetrical division.

9. In the case of small strains the volume increase is
— S\7<7, and this is called the cubic dilatation. If it
vanishes, the strain takes place with no change of volume,
that is, with no change of density. A strain of this char-
acter is called a transversal strain. There is a vector
potential from which a can be derived by the formula

a = VVt, SVt = 0.

There is no scalar potential since we do not generally have
also VVo- = 0. Indeed we have

2e = VV<r = WVVr = V2r - VSVr = V2r.

DEFORMABLE BODIES 259

This would give us the integral

\t = \irfffejr-dv.

The integration is over the entire body.

This strain is called transverse because in case we have a
a function of a single projection of p, on a given line, say a,
so that

a = afvx + /3f2-x + yf3-x,
SVo- = — /i = 0, fi = constant,

and all points are moved in this direction like those of a
rigid body. We may therefore take the constant equal to
zero, and /i = 0, so that

Saa = 0.

Hence every displacement is perpendicular to the line a.

10. When V\/a = 0, we call the strain longitudinal; for,
giving <j the same expression as in § 9, we see that we have

Wa =0 = 7/2' - fifs', and /2 = 0 = /3,
Vaa = 0.

Hence we have all the strain parallel to a.

11. In case the cubical dilatation iSVo" = 0, the strain
is purely of a shearing character, and if the curl VVv = 0,
the strain is purely of a dilatational character. Since any
vector a can be separated into a solenoidaJ and a lamellar
part in an infinity of ways, it is always possible to separate
the strain into two parts, one of dilatation only, the other
of shear only.

If we write a = VP + V\/t, then we can find P and r
in one way from the integrals

P = lir.ffSS<T'VTp-W,
r = - \TT'fffVa'VTp-l-dvf, p = p' - Pc.

260 VECTOR CALCULUS

The integrations extend throughout the body displaced.
This method of resolution is not always successful, and
other formulae must be used. (Duhem, Jour, des Math.,
1900.)

12. The components are not functionally independent,
but are subject to a set of relations due to Saint Venant.
These relations are obvious in the quaternion form, equiva-
lent to six scalar equations. The equation is

V-V<PoVV() = 0, if <p=SQV-<r,

where both V's operate on <p0. The equation is, further-
more, the necessary and sufficient condition that any linear
vector function <p can represent a strain. The problem of
finding the vector a when <p is a given linear and vector
function of p consists in inverting the equation

<p = — S() V -cr. (Kirchkoff, Mechanik, Vorlesung 27.)

It is evident that if we operate upon dp, we have

<pdp = do.

Hence the problem reduces to the integration of a set of
differential equations of the ordinary type.

EXAMPLES

(1). If (p = VeQ, we have or = Vep. Prove Saint Venant's
equations.

(2). If <p~ p-lV{)p-\ then a = Up. Prove Saint Ven-
ant's equations.

13. In general when we do not have small strains, we
must modify the. preceding theory somewhat. The dis-
placement will change the differential element dp into

dpi = dp — SdpV-<r.

The strain is characterized when we know the ratio of the
two differential elements and this we may find by squaring

DEFORMABLE BODIES 261

so as to arrive at the tensor

(dpi)* = Sdp[l - 2vSa + V'S(r'(T"SV"]dp-

The function in the brackets is the general strain function,
which we will represent by <£. It is easily clear that if
<p = — SQV'<r then

* = (1 + <p)(l + <p') = (1 + *>)(1 + <p)'.

Of course $ is self-conjugate. Its components Sa&fi are
also called components of strain. If <p is infinitesimal, we
may substitute (1 + 2<po) for <£.

The cubical dilatation is now found by subtracting 1 from

SdipidhpidtPi/Sdipdtpdtp = m3(l + <p) = 1 + A.

Evidently (1 + A)2 = m3($). The alteration in the
angle of two elements is found from

- suq. + <p)\u(i<p)y.

If angles are not altered between the infinitesimal elements,
the transformation is conformal, or isogonal. In such case

Eti&k' = s2\ys\$\s\'$y.

For example, if <p = VaQ,

sua + <p)\Q. + <p)v = sxx',

when Sa\ = 0 = Sa\'.

14. This part of the subject leads us into the theory of
infinitesimal transformations, and is too extensive to be
treated here.

On Discontinuities

15. If the function <j is continuous throughout a body,
it may happen that its convergence or its curl may be dis-
continuous. The consideration of such discontinuities is

262 VECTOR CALCULUS

usually given at length in a discussion of the potential
functions. Here we need only the elements of the theory.
We make use of the following general theorem from analysis.

Lemma. If a function is continuous on one side of a sur-
face for all points not actually on the surface in question, and
if, as we approach the surface by each and every path leading
up to a point P, the gradient of the function, or its directional
derivatives approach one and the same limit for all the paths;
then the differential of this function along a path lying on the
surface is also given by the usual formula,

— SdpV -q = dq, dp being on the surface.

[Hadamard, Lecons sur la propagation des ondes, etc.,
p. 84, Painleve, Ann. Ecole Normale, 1887, Part 1, ch. 2,
no. 2.]

In the case of a vector a which has the same value on
each side of a surface, which is the value on the surface,
and is the limiting value as the surface is approached, at
all points of the surface, we have on one side of the surface

da — — Sdp\7 •& = <pidp.

On the opposite side

da = — Sdp\7 -<r = <p2dp.

If now these two do not agree, but there is a discontinuity
in <p, so that <p2 — tp\ is finite as the two paths are made
to approach the surface, then designating the fluctuation or
saltus of a function by the notation [], we have in the limit

[da] = (<p2 — <Pi)dp = [<p]dp.

But since a does not vary abruptly, [da] along the surface
is zero, hence for dp on the surface

[<p]dp = 0,

DEFORMABLE BODIES 263

and therefore

M = — vSv>

where v is the unit normal, \x a given vector. That is to
say, we have for the transition of the surface

[S()V-a] = »Sv.
Whence

[SVcr] = Spix,
[W<t] = Vvix.

These are conditions of compatibility of the surface of dis-
continuities and the discontinuity; or identical conditions,
under which the discontinuities can actually have the sur-
face for their distribution.

16. If *S/x^ = 0, then [S Vo"] = 0, and the cubic dilatation
is continuous.

Since Svvjjl = 0 = Sv[V\7<t] = [SpV<t], the normal com-
ponent of the curl of a is continuous, and the discontinuity
is confined to the tangential component. Likewise

Sfivn = 0 = [S/xVo-],

and the component along ijl is continuous. Hence V\7(r
can be discontinuous only normal to the plane of /jl, v.

17. In case a itself is discontinuous, the normal com-
ponent of a as it passes the surface of discontinuity cannot
be discontinuous without tearing the surface in two. Hence
the discontinuity is purely tangential. It can be related
to the curl of a as follows. . .

Consider a line on the surface, of infinitesimal length, and
an infinitesimal rectangle normal to the surface, and let
the value of a at the two upper points differ only infinites-
imally, as likewise at the two lower points, but the differ-
ence at the two right hand points or at the two left hand

264 VECTOR CALCULUS .

points be finite, so that a has a discontinuity in going
through the surface equal to [a]. Then

fSbpa = ffSK(AWa)

around the rectangle, when k is normal to the rectangle.
But the four parts on the left for the four sides give simply

Sid*},

where 8p is a horizontal side and equal to V-wTSp. Hence
we have for every k tangential to the surface

SkV v[a] - Sk Urn (AW<r)IT8p.

Dropping all infinitesimals, we have

Vv[(t] = Lim AVVcr/Tdp.

Tangential discontinuities may therefore be considered
to be representable by a limiting value of the curl multi-
plied by an infinitesimal area, as if the surface of discon-
tinuity were the locus of the axial lines of an infinity of small
rotations which enable one space to roll upon the other.
The expression \[<j] is the strength of this sheet.

A strain is not irrotational unless such surfaces of dis-
continuity are absent. But we have shown above that a
continuous strain may imply certain surfaces of discon-
tinuity in its derivatives of some order. If V\7cr = 0,
everywhere, then Vv[u] = 0, and such discontinuity as
exists is parallel to v.

The derivation above applies to any case, and we may
say that if a field is irrotational, any discontinuity it pos-
sesses must be normal to the surface of discontinuity.

Integrating in the same way over the surface of a small
box, we would have

ffSv[<r]ds = SV<T'V,

DEFORMABLE BODIES 265

where v is the infinitesimal volume. But this gives

Sv[a] = vSV (r/surface.

If then $Vo" = 0 everywhere, the discontinuity of a is
normal to the normal, that is, it is purely tangential. These
theorems will be useful in the study of electro-dynamics.

Kinematics of Displacements
18. In the case of a continuous displacement which takes
place in time we have as the vector a the velocity of a
moving particle, and if p is the vector from a fixed point
to the particle, then dp/dt = a. It is necessary to distin-
guish between the velocity of the particle and the local
velocity of the stream of particles as they pass a given fixed
point in the absolute space which is supposed to be sta-
tionary. The latter is designated by d/dt. Thus dcr/dt is
the local rate of change of the velocity at a certain point.
While da/dt is the rate of change of the velocity as we follow
the particle. It is easy to see that for any quaternion q
the actual time rate of change is

dq/dt = dq/dt — SaV -q.

We have thus the acceleration

da/dt = da/dt - SaV-cr = (d/dt + <p)a.

If the infinitesimal vector dp is considered to be displaced,
we have

bdp/dt = - S5pV'(r.

Since the rotation is \V\7a dt, the angular velocity of turn
of the particle to which dp is attached is |FVo". This is
the vortex velocity. Likewise the velocity of cubic dilata-
tion is — S\/a.

The rate of change of an infinitesimal volume dv as it
18

266 VECTOR CALCULUS

moves along is

— SV<T'dv.

The equation of continuity is d(cdv) = 0, where c is the
density, or

dc/dt + c{- SV<r) = 0.

That is, we have for a medium of constant mass

dc/dt = cSVv-

That is, the density at a moving point has a rate of change
per second equal to the density times the convergence of
the velocity.

It may also be written easily

dc/dt = SVW.

This means that at a fixed point the velocity of increase in
density is equal to the convergence of the momentum per
cubic centimeter.

19. When FVo" = 0, the motion is irrotational, or dila-
tational, and we may put a = VP, where now P is a veloc-
ity-potential, which may be monodromic or polydromic.
When SVcr = 0, the motion is solenoidal or circuital, and
we may write a = VVr where &Vr = 0. r is the vector
potential of velocity. The lines e = \V\7<r become in this
case the concentration of Jr. The lines of a are the vortex
lines of r, and the lines of e are the vortex lines of a.

20. If a is continuous, and the equation of a surface of
discontinuity of the gradient dyadic of a and of a' is / = 0,
where now a is a displacement and a' is da/dt the velocity,
we have certain conditions of kinematic compatibility.
These were given by Christoffel in 1877-8 and are found as
follows. We have

M = o, [_0ov«W<-jtfi>

DEFORMABLE BODIES 267

in the case in which the time t is not involved; and for a
moving surface in which / is a function of t as well as of p,
we would have

[-SOV-<r]=-»SUvfO,

[" S Tt V<T] = " mS i Uvf= Mf /*V/=M=-Gm.

This gives us the discontinuity in the time rate of change of
the displacement of a point as it passes from one side to the
other of the moving surface. The equation of the surface
as it moves being /(/>, t), we have in the normal direction

- SdpV-f+dtf = 0,

that is, since dp is now Uvfdn, dn/dt — — f'/T\/f = G,
where / ' is the derivative of / as to t alone. In words,
at any point on the instantaneous position of the moving
surface the rate of outward motion of the point of the
surface coinciding with the fixed point in space is
G = —f'jTS/f. The moving surface of discontinuity is
called a wave and G the rate of propagation of the wave at
the given point. We may now read the condition of com-
patibility above in these words: the abrupt change in the
displacement velocity is given by a definite vector p. at
each point multiplied by the negative rate of propagation
of the wave of displacement, that is, if G is the rate of
propagation,

[o-'j = - Gp, and [SVff] = - SpVvf = - S/iv.

21. The preceding theorem becomes general for discon-
tinuities of any order in the following way. Let the func-
tion a and all its derivatives be continuous down to the
(n — l)th, then we can write

[SQiV'SQzV — S0*-iV-*]«0,

268 VECTOR CALCULUS

whence, differentiating along the surface of discontinuity
as before, we find in precisely the same manner

• [S()iV • • • SO. V •*) = nSOiUvfSihUvf • • • SQnUvf,

since at a given point on the fixed surface V/ is constant.
And if we insert dp/dt in m parentheses (m < ri), we
shall have, since the surface is moving,

= - »G»SQiUvf • • • S0n-mUvf(-l)m.

In particular for m — 2 = n, we have

W) = mG2,

which is the discontinuity in the acceleration of the dis-
placement.

If m = 1, n = 2,

[SOW] = - nGSQUVf.

From this we derive easily

[SVff'l = - GS»Uvf= - GSfxp.
[W<r'] = - GVfxUvf= ~ GVfip.

22. The nth. derivatives of Saa are

[S()iV • • • SQnV-Saa] = SQiUvf • • ■ SQnUVfSap.

If then we hold the surface fixed and consider a certain
point, the discontinuity in the nth derivative of the ratio
of two values of the infinitesimal volume which has two
perpendicular directions on the surface and the third along .
the normal will be given by the formula

SQiUvf ■ ■ ■ SOnllVfSnUvf.
In case we have a material substance that has mass and

DEFORMABLE BODIES 269

density and of which the mass remains fixed, we have

c/cq = volo/voi,
log c — log Co = log v0 — log V,
V log c = — V log v/v0 = — Vo/v-V(clvo).

Therefore from the formula above we have since v0/v = 1
in the limit

[SOiV • • • S()nV log c] = SQiUVf • • • SQnUvfSfjiUvf.

In particular for the case of discontinuities of order two-
we have

[Vlogc]= UvfSfiUvf.

23. These theorems may be extended to the case in which
the medium is in motion as well as the wave of discontinuity.

Stress

24. In any body the stress at a given point is given as a
tension or a pressure which is exerted from some source
across an infinitesimal area situated at the point. The
stress real y consists of two opposing actions, being taken
as positive if a tension, negative if a pressure. It is as-
sumed that the stress taken all over the surface of an
infinitesimal closed solid in the body will be a system of
forces in equilibrium, to terms of the first order. This is
equivalent to assuming that the stress on any infinitesimal
portion of the surface is a linear function of the normal,
that is

6 = ZVv.

25. We have therefore for any infinitesimal portion of
space inside the body

ffQdA = ffZdv = 0.

But by Green's theorem this is equal to the integral through

270 VECTOR CALCULUS

the infinitesimal space J J VHV = 0. Hence SV = 0.
In this equation S is a function of p, and V differentiates S.

26. In case the portion of space integrated over or
through is not infinitesimal, this equation (in which S is
no longer a constant function) remains true if there is
equilibrium; and if there are external forces that produce
equilibrium, say £ per unit volume, then the density being
c, we have

SV + c£ = 0
for every point.

In case there is a small motion, we have

EV + c£ = co".

27. Returning to the infinitesimal space considered, we
see that the moment as to the origin of the stress on a
portion of the boundary will be VpSJJv and the total
moment which must vanish, considering S as constant, is

ffVpZdv = fffVpttdv,
hence

FpHv = 0 = €(S).

We see therefore that S is self-conjugate.

EXAMPLES

(1). Purely normal stress, hydrostatic stress. In this case
S is of the form pS = gp, where g is + for tension, — for
pressure, and is a function of p (scalar, of course).

(2). Simple tension or pressure.

H = — paSa.
(3). Shearing stress.

H = - p(aSp + PSa),
|S not parallel to a.

DEFORMABLE BODIES 271

(4). Plane stress.

8 - giaSa + g2(3S(3.
(5). Maxwell's electrostatic stress.

H= l/87r-FvP()VP,

where P is the potential.

28. The quadric Spap = — C is called the stress quadric.
Its principal axes give the direction of the principal stresses.
Since Sp is the direction of the normal we may arrive at a
graphical understanding of the stress by passing planes
through the center, and to each construct the conjugate
diameter. This will give the direction of the stress, and
since Tap is inversely proportional to the perpendicular
from the origin on the tangent plane at p, if we lay off on
the conjugate diameter distances inversely as the per-
pendiculars, we shall have the vector representation of the
stress. When the diameter is normal to its conjugate plane,
there will be no component of the corresponding vector
that is parallel to the plane, that is, no tangential stress.
Such planes will be the principal planes of the stress.

It is evident that a stress is completely known when the
self-conjugate linear vector function H is known, which
depends therefore upon six parameters. We shall speak,
then, of the stress H, since H represents it. This proposi-
tion is sometimes stated as follows: stress is not a vector
but a dyadic (tensor). From this point of view the six
components of the stress are taken as the coordinates of a
vector in six-dimensional space. These components in the
quaternion notation are, for a, (3, y, a trirectangular system,

- SaXa, - S/3E0, - SyZy, - Saafi = - S(3Za,
- SpEy = - SyZp, - SyZa = - Sa3y.

272 VECTOR CALCULUS

That is,

Xx Y y Zt, Xy = YXi Y t = Zy, Zix ~ X 2.

It is easy to see now that certain combinations of these
component stresses are invariant. Thus we have at once
the three invariants mi, m2, m3, which are

Xx ~r* Yy~\~ %zt YyZz -f- ZgXx ~r XxYy — Yz — Zx — Xy ,
XtY yZ z -\- ZXyY ZZX XXY z Y yZx ZzXy .

For any three perpendicular planes these are invariant.

EXERCISE

What are the principal stresses and principal planes of the five ex-
amples given above?

29. Returning to the equation of a small displacement,
we may write it

er" = i + <TlEV.

Hence the time rate of storage or dissipation of energy is

W'=- fffSa'Zvdv.

The other terms of the kinetic energy are not due to storage
of energy.

Now we have an experimental law due to Hooke which in
its full statement is to the effect that the stress dyadic is a
linear function of the strain dyadic. The latter was shown
to be

<Po= -^S()V-<7+ V&rOJ.

The law of Hooke then amounts to saying that S is a linear
function of a and V where V operates upon a, and owing
to the self-conjugate character of <p, we must be able to
interchange V and a, that is,

S = 6[(), V, a}.

DEFORMABLE BODIES 273

First, it follows that if the strain <p0 is multiplied by a
variable parameter x, that the stress will be multiplied by
the same parameter. We have then for a parametric change
of this kind which we may suppose to take place in a alone
a' = ax' . Hence for a gradually increasing a, we would
have

W = - xx'fffSaSVdv,

w = - iyyy&rEv &%

if x runs from 0 to 1. This gives an expression for the
energy if it is stored in this special manner. If the work is
a function of the strain alone and not dependent upon the
way in which it is brought about, W is called an energy-
function. It is thus seen to be a quadratic function of the
strain. In case there is an energy function, we have for
two strain functions due to the displacements crlf a2

Si = e[(), en, Vi], H2 = G[(), o-2, v2]-

The stored energy for the two displacements must be the
same either way we arrange the displacements, hence we
have

So-2e3[V3, *i» Vi] = (Scr1e4[V*i <r2, V2],

where the subscripts 3, 4 merely indicate upon what V acts.
This is equivalent to saying that so far as vector function
is concerned, in the form

SaG[(3, 7, 5]

we can interchange a, (3 and y, 5. Since S is self-conjugate,
0 is self-conjugate, and we can interchange a and (3. From
the nature of the strain function we can interchange y, 8.
Of course, in the forms above we cannot interchange the
effect of the differentiations.

274 VECTOR CALCULUS

We have in this way arrived at six linear vector functions

<P\l <f22 <P32 <f23 <fn <Pl2>

wherein we can interchange the subscripts, and where
<Pn = 0[Q,a,a] ••• ^23= 6[(),ft7] v\,

a /3 7 being a trirectangular system of unit vectors. We
have further a system of thirty-six constituents Cmu cni2,
• • • where

Cim = — Sa<pn<x, C1112 = — Sa<pn<x, • • •,

each of the six functions having six constituents. These
are the 36 elastic constants. If there is an energy function,
they reduce in number to only 21, for we must be able to
interchange the first pair of numbers with the last pair.
There are thus left

3 forms emu 6 of em%, 3 of Cim, 3 of C1212, 3 of C2311, 3 of 02m.

In theories of elasticity based upon a molecular theory
and action at a distance six other relations are added to
these reducing the number of elastic constants to 15. These
relations are equivalent to an interchange of the second
and third subscript in each form, thus Cim = Ci2is- These
are usually called Cauchy's relations, but are not commonly
used. (See Love, Elasticity, Chap. III.)

Remembering the strain function <p0, we can interpret
these coefficients with no difficulty, for we have

— SaipoCXj • fty,

the stress dyadic due to the strain component — Sa&oaj,
where a;, a; are any two of the three a, (3, y. cijki is the
component of the stress across a plane normal to otj in the
direction at due to the strain component — Sak<Poai-

DEFORMABLE BODIES 275

EXAMPLES

(1). If Sij = — Soti<pocxj, show that we have for the energy
function

W = ^CnnSn + 2cii22SnS22 + i^c12i2s122

+ 201223^12^23 + SCni2*ll*l2 + 2Cii2SSnS23.

(2). When there is a plane of symmetry, say in the direc-
tion normal to 7, all constants that involve 7 an odd number
of times vanish, for the solid is unchanged by reflection in
this plane. Only thirteen remain. If there are two per-
pendicular planes of symmetry, normal to (3, y, the only
constants left are of the types

ClUli C1122, Ci212j

the plane normal to a is thus a plane of symmetry also.
There are nine constants. This last case is that of tesseral
crystals.

(3). If the constants are not altered by a change of a into
— a, (3 into — (3, as by rotation about 7 through a straight
angle, then the plane normal to 7 is a plane of symmetry.

(4). Discuss the effect of rotation about 7 through other
angles.

(5). When the energy function exists we have

0(X, fi, v) - 90*, X, v) = - VvQV\\x, where 6' = 6.

30. A body is said to be isotropic as to elasticity when the
elastic constants are not dependent upon directions in the
body. In such case the energy function is invariant under
orthogonal transformation. It must, therefore, be a function
of the three invariants of <po, i»i, ra2, m3. The last is of
third degree, while the energy function is a quadratic
and therefore can be only of the form

W = - Pmi + Am? + Bm2.

276 VECTOR CALCULUS

P is zero except for gases and is then positive. The con-
stant A refers to resistance to compression, and is positive.
B is a constant belonging to solids.

The form given the quadratic terms by Helmholtz is

Amx2 + Bm2 = iHml2 + £C[2mi2 - 6m2].

The [] is the sum of the squares of the differences of the
latent roots of <po. The constant H refers to changes of
volume without change of form, and in such change it is the
whole energy, for if there is no change of form, the roots
are all equal and the other term is zero. C refers to changes
of form without change of volume, since it vanishes if the
roots are equal and is the whole energy if there is no cubical
expansion m\. For perfect fluids C = 0.
The form given by Kirchoff is

Km^tpo2) + Kdrm2.
From which we have

B-C = 2KB, 3C = 2K, H= 2K(d + |), C = \K.
We may write for solids, liquids, and gases

W = Rdm? + Kmifao*) - Pmx.
Later notation gives 2K6 = X, K = /x, that is,

W = |Xmi2 + iirriiicpo2) — Pm\.

The constants X, \x are the two independent constants of
isotropic bodies.

We now have for the stress function in terms of the strain
function

S = Xrai -f- 2/i^o.

EXAMPLES
(1). In the case of a simple dilatation we know S = p

DEFORMABLE BODIES 277

and we have for <po

<Po= - JOSOV-ap + ASapQ) = a().

Substituting in the equation above, we have

()p m X(3o) + 2Mo().

The cubical dilatation is thus

3a = p/(X + |m) = p/»,

where A: is called the modulus of cubical compression.
(2). For a simple shear

<p, = - a/2-[aSPQ + g&xOL ™i = 0>
S = - a/z[«<Sj8() + 0&*()].

If the tangential stress is T, then T = a/j,. M is the shear
modulus or simple rigidity.

(3). If a prism of any form is subject to tension T uniform
over its plane ends, and no lateral traction, we have

S = - afSaQ - Xm + 2n<p0.

From this equation, taking the first scalar invariant of
both sides,

T = 3mA + 2muh
so that

rrn= T/(3\+2fi).

Substituting, we have

2/i v 2ju(3X +2ju)

We write now E = /x(3X + 2/x)/(X + /x)> the quotient of a
simple longitudinal tension by the stretch produced, and
called Young's modulus. Also we set

s = X/(2X + 2/x), Poisson's ratio,

278 VECTOR CALCULUS

the ratio of the lateral contraction to the longitudinal
stretch.

It is clear that if any two of the three moduli are known,
the other may be found. We have

X = E/[(l + *)(1 - 2*), M - \Ej(X + *),
k - IE/(1 - 2s).

In terms of E and s we have

t»i(S)'«

po

-m*

E

(4). If | < s, k < 0, and the material would expand under
pressure. If s < — 1, W would not be positive.

(5). If Cauchy's relations hold, s = \ and X = /x. For
numerical values of the moduli see texts such as Love,
Elasticity.

31. Bodies that are not isotropic are called aelotropic.
For discussion of the cases and definitions of the moduli,
see texts on elasticity.

32. There is still the problem of finding a from cp0 after
the latter has been found from S. This problem we can
solve as follows:

<t = <tq-\- fp^da = (To — J£<rS Vdp, where V acts on a
= o-o + fgW&P ~ hVdpVVv]
= *o + fgWdp ~ WiPi ~ p)VVd<r

-d-V(Pl- p)VV<r]
= <to- Wifii ~ Po)VVao+ fPPModp

-iVQ>i-p)VvM

= <ro- \Vifii - Po)VV<ro + fPSl[<Podp

- V(Pl- p)W<Po'dp].

We are thus able to express a at any point pi in terms of the

DEFORMABLE BODIES 279

values at p0 of cr, VVc, and the values along the path of
integration of <p0 and FV^oO-

EXAMPLES

(1). Let us consider a cylinder or prism which is vertical
with horizontal ends, the upper being cemented to a hori-
zontal plane. Then we have the value of

% = — gcySypSyQ, y vertical unit,

where the origin is at the center of the lower base. The
conditions of equilibrium are

S V + c£ =0, or c{ * - gey, J = - gy.

That is, the condition is realizable by a cylinder hanging
under its own weight. The tension at the top surface is
gel where I is the length.

Solving for the strain, we have

Let a = gcs/E, b = gc(l + s)IE, and note that

FWoO = - aVy() - bVyySyQ = - aVy().

The integral is thus

°"o — hV(fti — p0)e0

+ fp'oiaSyp-dp + bySypSydp + aV(Pl - p)Vydp]
= (To— \V(p\ — po)e0

+ Jl?[aSyp-8p + bySypSydp

+« VpiVydp — adpSyp + aySpdp]
= a0 — W{pi — p0)e0 + HbyS2yp

+ aVPlVyp + haypX,

the differential being exact. This gives us as the value of
a at pi,

280 VECTOR CALCULUS

*l « f + V(pi - p0)(ieo + aVypo) + iaFprypi

-f \byS2yp> f > « constants.

Substituting a and 6, and constructing

<Po = - J[S()V-«r+ V&r()],

we easily verify. If the cylinder does not rotate, we may
omit the second term and if the upper base does not move
laterally, then the vector f reduces to — ^gcP/E-y, and
we have

' = - hgcP/E-y + gcs/2E-Vpyp + gc(l + s)/2E-yS2py.

A plane cross-section of the cylinder is distorted into a
paraboloid of revolution about the axis and the sections
shrink laterally by distances proportional to their distances
from the free end.

(2). If a cylinder of length 21 is immersed in a fluid of
density c', its own density being c, the upper end fixed, p
the pressure of the fluid at the center of gravity, then we
have the stress given by

H = - (p + gc'Syp) - g(c - c')(l - Syp)ySy,

whence calculating <p0, we have

<p0 = 1/E-l- (p + gc'Syp)(- 1 + 2*) - gs(c - c')

X (1 - Syp)] - ySy[g(c - c')(l - Syp)l + s)]/E.
And

a = f + Vdp + p[(- 1 + 2*)p - ^/(c - cO.
- Spyg[ce - s(c + c')]/E

+ 7lh(c ~ c')(l + s)(l - Syp)2 '

+ hgp2W-s(c+c')]/E.

(3). What does the preceding reduce to if c = a'? Solve
also directly.

DEFORMABLE BODIES 281

(4). If a circular bar has its axis parallel to y, and the
only stress is a traction at each end, equivalent to couples
of moment \ira*pt, about the axis of y, a being the radius,
that is, a round bar held twisted by opposing couples, we
have

S = - lidfySnO + VpySyQ),
<Po= - HiySpyO + VpySyQ],
a = tVpySyp.

Any section is turned in its own plane through the angle
— tSyp. t is the angular twist per centimeter.

(5). The next example is of considerable importance, as
it is that of a bar bent by couples. The equations are

g = - E/R-Sap-ySyQ,

Po - - (1 + s)/R-Sap-ySyQ ~ s/R-Sap-Q,
a = iR-i-al&yp + sS2ap - sS2yap]

+ sR~1yaS(3pSap — R~1ySapSyp.

If the body is a cylinder or prism of any shape with the
axis y horizontal, there is no body force nor traction on the
perimeter. The resultant traction across any section is

ff- EjR-SapdA,

which will equal zero if the origin is on the line of centroids
of the sections in the normal state, that is, the neutral axis.
Thus the bar is stressed only by the tractions. at its terminal
sections, the traction across any section being equivalent
to a couple.

The couple becomes one with axis (3 = ya and value
EI/R, where 7 is the moment of inertia about an axis
through the centroid parallel to (3. The line of centroids
is displaced according to the law

- Saa = iR-'S'-yp,

19

2N2 VECTOR CALCULUS

so that it is approximately the arc of a circle of radius R.
The strain-energy function is \ER~2-S2ap, and the potential
energy per unit length %EI/R2.

For further discussion see Love, p. 127 et seq.

(6). When E = - E-Syp-OQ, where dy = 0, and 6 = 0',
and a may not be a unit vector, show that

<Po = ~ (1 + 8)Syp-6Q + sSyp-mi(0),
a = (1 + 8)tiSp6p - OpSpy] + mi«[- \yp2 + pSpy].

See Love, pp. 129-130.

33. We recur now to the equation of equilibrium

EV + cf - 0.

In this we substitute the value of

H = Xmi + 2/^o = - XSVo- - (o-/S() V + V&r()),

whence

XV*SVcr + m W + n\/SS7<r - cf = 0,
or

(X + M) VSVo- + MV2c - c? = 0,

or equally since

VV = VSVa + VVVa,
(X + 2M) V»SVcr + fiVVVcr - c£ = 0.

This is the equation of equilibrium when the displacement
and the force £ are given. In the case of small motion we
insert on the right side instead of 0, — ca". The traction
across a plane of normal v is

— (X + iJ,)vSVcr — pV\Jvv,

where v is constant. Operating on the equilibrium equa-
tion by *SV(), we see that

(X+2/z)V2SV<r-oSv£= 0.

DEFORMABLE BODIES 283

If then there are no body forces £ or if the forces £ are
derivable from a force-function P and V2P = 0 throughout
the body, we see that

SVa

is a harmonic function. Since rai(E) = Skmi, we see that
mi(H) is also harmonic.
Again we have

(X + m)V#Vo- = - MVV,

whence we can construct the operators

(X + /xj V£v()£V<r - - mV2V&j - - Mvvsv().
and adding the two,

2(X + M)VSvSV(r() - - mV2(^V() + V&r())
Now we have

g = - \SVct - m(^V() + V-ScrO),
and since S\7<r is harmonic

V2H = - /xV2(^V() + V&r()) = 2(X + /*) ViS ViSVcrO
2(X + M)

3&
or

V#V£V<7() = (1 + s^VSvSVtrQ.

V2H = ^- ViSvifiO.

This relation is due to Beltrami, R. A. L. R., (5) 1 (1892).

EXAMPLE
Maxwell's stress system cannot occur in a solid body
which is isotropic, free from the action of body forces, and
slightly strained from a state of no stress, since we have
-Wil(E) = 1/8tt-(vP)2,

284 VECTOR CALCULUS

which is not harmonic. (Minchin Statics, 3d ed. (1886),
vol. 12, ch. 18.)

34. We consider now the problem of vibrations of a solid
under no body forces, the body being either isotropic or
aeolotropic.

The equation of vibrations is

c<r" = 6( V, V, <r), where S = 6[(), V, <r] as before, and

a is a function of both t and p. If the vector co represents
the direction and the magnitude of the wave-front, the
equation of a plane-wave will be

u = t — Sp/co,

since this represents a variable plane moving along its
own normal with velocity w. By definition of a wave-front
the displacement from the mean position is at any instant
the same at every point. That is, a is a function of u
and t, hence

Vo" = — VSp/ooda/du = uT^a/du,

and any homogeneous function of V as/(V) gives

/V-<r = f{oTl)dn(rldun,

where n is the degree of /.

The equation above for wave-motion then is

cv" = e[oj-\ or1, d2a/du2].

If the wave is permanent, a involves t only through u and
if the vibration is harmonic of frequency p,

<r" = du2a/d2 = - fa.
Therefore

e[Uu, Uw, a] = ctrT*u.

Hence for a plane wave propagated in the direction Uoj

DEFORMABLE BODIES 285

the vibration is parallel to one of the invariant lines of the
function

e[U<a, Uco, ()].

The velocity is the square root of the quotient of the latent
root corresponding, by the density. There may be three
plane-polarized waves propagated in the same direction
with different velocities. The wave- velocity surface is
determined by the equation

S[e(w-\ co"1, a) - ca][e(u-\ co"1, (3 - cjSHeC&T1, co"1, y] = 0,

that is, by the cubic of Q[Uu, Uu, ()].

If there is an energy function, Q[Uu, Uu, ()] is self-
conjugate as may easily be seen. In such case the invariant
lines are perpendicular, that is, the three directions of
vibration, 0U 62, 03, for any direction of propagation are
mutually trirectangular. Since W is essentially positive,
the roots are positive, and there are thus three real velocities
in any direction.

If g is a repeated root, there is an invariant plane of
indeterminate lines and the condition for such is

V[e(«T\ to"1, a) - ca][e(^-\ co"1, 0) - cfi] = 0,

a and /3 arbitrary. There is a finite number of solutions to
this vector equation, giving co, and these give Hamilton's
internal conical refraction. The vectors terminate at
double points of the wave-velocity surface.

The index-surface of MacCullagh, that is, Hamilton's
wave-slowness surface, is given by

5[0(p, p, a) - ca][G(p, p, (3) - cj8][0(p, p, 7) ~ ey] = 0,

a, jS, 7 arbitrary, which is the inverse of the wave-velocity
surface, p is the current vector of the surface, just as co
for the other surface, the equation being formed by setting

286 VECTOR CALCULUS

p = — a>-1. The wave-surface, or surface of ray- velocity,
is the envelope of Sp/o) = 1, or Spp = — 1, where
/x = — w_1. The condition is that given by the equations
of the two other surfaces. It is the reciprocal of the index
surface with respect to the unit sphere p2 = — 1, or the
envelope of the plane wave-fronts in unit time after passing
the origin, or the wave of the vibration propagated from the
origin in unit time. The vectors p that satisfy its equation
are in magnitude and direction the ray- velocities. When
there is an energy function, this ray-velocity is found
easily, as follows:

The wave-surface is the result of eliminating between

0(/x, p, a) = ca,

Q(dp, p, a) + 0(ju, dp, a) + 0(ju, /x, da) = cdcr,
Sup = - 1-Spdfi= 0.

From the second equation

2SdfxG(<T, a, n) + SdaOiii, fi, a) = cSadX,

or by the equations

Sdp.e(<r, a, /x) = 0.

Hence as dfi is perpendicular to p, we have

G(<r, <r, p) = xp.

Operate by Sp and substitute the value of x,

Q(U<t, Ua, p) = cp.

This equation with 6(p,p, a) = ca gives all the relations
between the three vectors. See Joly, p. 247 et seq.

CHAPTER XI
HYDRODYNAMICS

1. Liquids and gases may be considered under the com-
mon name of fluids. By definition, a perfect fluid as dis-
tinguished from a viscous fluid has the property that its
state of stress in motion or when stationary can be con-
sidered to be an operator which has three equal roots and
all lines invariant, thus

E = -p(),

where p is positive, that is, a pressure, or S = —p. If the
density is c, we have, when there are external forces and
motion, the fundamental equation of hydrodynamics

<r" = J - c~lVp.

In the case of viscous fluids we have to return to the
general equation

c (*" - {) « - Vp - (X + m) VSV o- -mW.

2. When there is equilibrium

Vp = c£.

If the external forces may be derived from a force function,
P, we have Vp = cVP, hence — SdpVp = — ScdpVP,
or dp = cdP for all directions. That is, any infinitesimal
variation of the pressure is equal to the density into the
infinitesimal variation of the force function. In order that
there may be equilibrium under the forces that reduce to
£, we must have £ subject to a condition, for from Vp = c£,
we have V2p = Vc£ + cV£, whence ££V£ = 0, and
VV% = F£Vlogc.

287

288 VECTOR CALCULUS

If £ = VP, the condition is, of course, satisfied, and
from the last equation we see that £ is parallel to Vc, that
is to say, £ is normal to the isopycnic surface at the point,
or the levels of the force function are the isopycnic surfaces.
The equation Vp = c£ states that £ is also a normal of the
isobaric surfaces. In other words, in equilibrium the iso-
baric surfaces, the isopycnic surfaces, and the isosteric sur-
faces are geometrically the same. However, it is to be
noted that if a set of levels be drawn for any one of the
three so that the values of the function represented differ
for the levels by a unit, that is, if unit sheets are constructed,
then the levels in the one case may not agree with the levels
in the other two cases in distribution.

The fundamental equation above may be read in words:
the pressure gradient is the force per unit volume. Specific
volume times pressure gradient is the force per unit mass.

We can also translate the differential statement into
words thus: the mean specific volume in an isobaric unit
sheet is the number of equipotential unit sheets that are in-
cluded in the isobaric unit sheet. The average density in an
equipotential unit sheet is the number of isobaric unit sheets
enclosed.

Since dp and dP are exact differentials, we have :

Under statical conditions the line integral of the force of
pressure per unit mass as well as the line integral of the force
from the force function per unit volume are independent of the
path of integration and thus depend only on the end points.

3. There is for every fluid a characteristic equation which
states a relation between the pressure, the density, and a
third variable which in the case of a gas may be the tempera-
ture, or in the case of a liquid like the sea, the salinity.
Thus the law of Gay-Lussac-Mariotte for a gas is

p = const -c (1+ t^t T) f°r constant volume.

HYDRODYNAMICS 289

The characteristic equation usually appears in the form
pa = RT, where in this case a is the specific volume, the
equation reading

dP = adp.
From this we have

dP = RTdp/p.

If T is connected with p by any law such as that given
above, we can substitute its value and integrate at once.
Or if T is connected with the force function P by an equa-
tion, we can integrate at once.

Example.
In the case of gravity and the atmosphere, suppose
that the temperature decreases uniformly with the equi-
potentials. Since we must in this case take P so that
VP will be negative, we have

dP = - RTdp/p, T = T0 - bP,
whence

dP = -dT/b, dT/T = Rbdp/p, T = T0(p/p0)bR.

Or again

dP/(T0 -bP)= - R dp/p, 1 - bP/To = (p/po)R.

We thus have the full solution of the problem, the initial
conditions being for mean sea-level, and in terms of a or
c as follows :

T= T0(p/p0)bR, a= a0(p/po)bR-\

p = b-iT0[i- (p/p0n

T= T0(l - bTo-'P), c= c0(l - To-'bP) »"1»"1-1,
p = Pod- To-'bP)*-1*-1.

Absolute zero would then be reached at a height where the

200 VECTOR CALCULUS

gravity potential would be

P = To/b,

and substituting we find c = 0, p = 0. If b is negative,
the fictive limit of the atmosphere is below sea-level. For
values of bR from oo to 1, for the latter value b = 0.00348
(that is, a temperature drop of 3.48° C. per 100 dynamic
meters of height), we have unstable equilibrium, since
from the equations above for c we have increasing density
upwards. The case bR = 1 is extreme; however, it is
mathematically interesting from the simplicity that re-
sults. Pressure and temperature would decrease uniformly
and we should have a homogeneous atmosphere. This
condition is unstable and the slightest displacement would
continue indefinitely. Values of bR less than 1 lead still
to unstable equilibrium, the state of indifferent equilibrium
occurring when the adiabatic cooling of an upward moving
mass of air brings its temperature to that of the new levels.
For dry air this occurs for bR = 0.2884 = (1.4053
- 1) /1.4053, or a fall of 1.0048° C. per dynamic hectom-
eter.

See Bjerknes, Dynamic Meteorology and Hydrography.

4. The equation when there is not equilibrium gives us

aVp — £ * — a".

Let £ = VP, and operate by V*V (), then

WaVp = - VV<r".

If we multiply by SUv and integrate over any surface nor-
mal to Up, we have

SfSUvWaVp = - ffSUvW" = - fSdpa".

The right-hand side is the circulation of the acceleration
or force per unit mass around any loop, the left-hand side

HYDRODYNAMICS 291

is the surface integral of WaVp over the area enclosed.
If then we suppose that in a drawing we represent the iso-
bars as lines, and the isosterics also as lines that cut these,
drawing a line for the level that bounds a unit sheet in each
case (and noticing that in equilibrium the lines do not in-
tersect), we shall have a set of curvilinear parallelograms
representing tubes. The circulation of the force per unit
mass around any boundary will then be the number of
parallelograms enclosed. It is to be noticed that the areas
must be counted positively and negatively, that is, the
number of tubes must be taken positive or negative, ac-
cording to whether Vfl, Vp, the two gradients, make a
positive or a negative angle with each other in the order as
written. This circulation of the force per unit mass may be
taken as a measure of the departure from equilibrium.
In the same way we find that if we draw the equipotentials
and the isopycnics, we shall have the number (algebraically
considered) of unit tubes in any area equal to the circula-
tion of the force per unit volume around the bounding
curve.

If we choose as boundary, for example, a vertical line, an
isobaric curve, a downward vertical, and an isobaric curve,
the number of isobaric-isosteric tubes enclosed gives the differ-
ence between the excess up one vertical of the cubic meters
per ton at the upper isobar over that at the lower isobar and
the corresponding excess for the other vertical. If the lines
are two verticals and two equipotentials, the number of
isopotential-isopycnic tubes is the difference of the two
excesses of pressure at the lower levels over pressure at the
upper levels. These are the circulations around the bound-
aries of the forces per unit mass or unit volume as the case
may be.

5. If we integrate the pressure over a closed space inside

292 VECTOR CALCULUS

the fluid, we have

ffyUvdA = fffVpdv = fffc&v.

But this latter integral is the total force on the volume
enclosed. This is Archimedes' principle, usually related
to a body immersed in water, in which case the statement
is that the resultant of all the pressure of the water upon
the immersed body is equal to the weight of the water dis-
placed. If we were to consider the resultant moment of
the normal pressures and the external forces, we would
arrive at an analogous statement. The field of force, how-
ever, need not be that due to gravity.

EXERCISE.

Consider the case of a field in which there is the vertical
force due to gravity and a horizontal force due to centrif-
ugal force of rotation.

6. We turn our attention now to moving fluids. A
small space containing fluid with one of its points at po
may be followed as it moves with the fluid, always con-
taining the same particles. It will usually be deformed in
shape. The position p of the particle initially at p0 will
be a function of p0 and of t, say

p = 0 (p0, t).

The particle initially at p0 + dp will at the same time t
arrive at the position

p + dip = 6 (p0 + dp, t) = p — SdipVo-p,

hence dip becomes at time t

— SdipVo'P = <pdipo.

It follows that the area Vdipd2p = V(pdip0(pd2po, and the

HYDRODYNAMICS 293

volume

— Sdipdtpdzp = — S(pdipo(pd2po<pdspo =

— SdipQd2Poddp0' ms((p) .

If the fluid has a constant mass, then we must have

cdv = Codvo, or cra3 = c0.

This is the equation of continuity in the Lagrangian form.
The reference of the motion to the time and the initial con-
figuration is usually called reference to the Lagrangian
variables.
7. Since

dp = — SdpVp = — S<pdp0Vp

= — Sdpo<p'Vp = — SdpoVoP,
VoP = <p'Vp = - VoSpV-p.

But the equations of motion are already given in the form

aVp = £ - p",

hence in terms of the variables po and t we have

aVop = <p'(p — p")-

This equation, the characteristic equation of the fluid

F(p, c, T) - 0,

and the equation of continuity, give us five scalar equations
expressing six numbers in terms of p0 and t. In order to
make any problem definite then, we must introduce a
further hypothesis. The two that are the most common
are

(1) The temperature is constant, if T is temperature,
or the salinity is constant, if T is salinity. In case both
variables come in, we must have two corresponding hypoth-
eses:

21)4 VECTOR CALCULUS

(2) The fluid is a gas subject to adiabatic change.
The relation of pressure to density in this case is usually
written

p = kcy.

y is the ratio of specific heat under constant pressure to
that under constant volume, as for example, for compressed
air, 7 - 1-408.

8. In the integrations we are obliged to pay attention
to two kinds of conditions, those due to the initial values of
the space occupied by the fluid at t = 0, the pressure p0
and density c0, or specific volume a0, at each point of the
fluid, and the initial velocities of the particles p0' at p0.
The other conditions are the boundary conditions during
the movement. As for example, consider a fluid enclosed
in a tank or in a pipe or conduit. The velocity in the
latter case must be tangent to the walls. If we have the
general case of a moving boundary for the fluid, then its
equation would be

/(P, t) = o.

If then p' is the velocity, we must have

- SdPVf+ (df/dt)dt = 0, or - £p'V/+ df/dt = 0.

If there is a free surface, then the pressure here must be
constant, as the pressure of the air. In order to have
various combinations of these conditions coexistent, it is
necessary sometimes to introduce discontinuities.

9. If we were in a balloon in perfect equilibrium moving
along with one and the same mass of air, the barograph
would register the varying pressures on this mass, the ther-
mograph the varying temperatures, and if there were a
velocitymeter, it would register the varying velocity of the
mass. From these records one could determine graphically
or numerically the rates of change of all these quantities as

HYDRODYNAMICS 295

they inhere in the same mass. That is, we would have the
values of

dp/dt, dT/dt, dp/dt.

These may be called the individual time-derivatives of the
quantities. As the balloon passed any fixed station the
readings of all the instruments would be the same as instru-
ments at the fixed stations. But the rates of change would
differ. The rates of change of these quantities at the same
station would be for a fixed p and a variable t, and could
be called the local time-derivatives, or partial derivatives.
They can be calculated from the registered readings. The
relation between the two is given by the equation

d/dt = d/dt - Sp'V.

Thus we have between the individual and the local values
the relations

The last equation gives us the individual acceleration
in terms of the local acceleration and the velocity. From
the fundamental equation we have

ovp = f - dp' let + w \p' = i - dp'ldt ~ *<j>%

where the function

0=-S()V-p', 0'=-VV(), 0o =

K-sovy-wo),

2e = FVp'.
This statement of the motion in terms of the coordinates of

296 VECTOR CALCULUS

any point and the time is the statement in terms of Eulers
variables.

Since near po, p = po + po'dt, we have the former
function <p at this point in the form

<p= - S()Vo-p = l + <ft(- S()V-p')0= 1 + d^atpo.

Whence

m3(<p) = 1 + dtmi(6) = 1 + dt{- SVp').

Since the initial point is any point, this equation holds for
any point and we have the equation of continuity in the
form

c - cdtSVp' = c0 = c0 + dt-dc/dt(l ~ dtSVp'),

or, dropping terms of second order,

dc/dt - cSVp' = 0.

This is the* equation of continuity in the Euler form.
If we use local values,

dc/dt- SV(cp') = 0.

That is, the local rate of change of the density is the con-
vergence of specific momentum. It is obvious that if the
fluid is incompressible, that is, if the density is constant,
then the velocity is solenoidal. If the specific volume at a
local station is constant, then the specific momentum is
solenoidal. If the medium is incompressible and homo-
geneous, then both velocity and specific momentum are
solenoidal vectors. It is clear also that in any case the
normal component of velocity must be continuous through
any surface, but specific momentum need not be. If any
boundary is stationary, then both velocity and specific
momentum are tangential to it.

HYDRODYNAMICS 297

In the atmosphere, which is compressible, specific mo-
mentum is solenoidal, but in the incompressible hydro-
sphere, both velocity and specific momentum are solenoidal.
Of course the specific volume of the air changes at a
station, but only slowly, so that the approximate statement
made is close enough for meteorological purposes.

If at any given instant we draw at every point a vector
in the direction of the velocity, these vectors will determine
the vector lines of the velocity which are called lines of
flow. These lines are not made up of the same particles
and if we were to mark a given set of particles at any time,
say by coloring them blue, then the configuration of the
blue particles would change from instant to instant as they
moved along. The trajectory of a blue particle is a stream
line. If the particles that pass a given point are all colored
red, then we would have a red line as a line of flow, only when
the condition of the motion is that called stationary. In
this case the line through the red particles would be the
streamline through the point. If the motion is not sta-
tionary, then after a time the red particles would form a
red filament that would be tangled up with several stream
lines.

10. In the case of meteorological observations the di-
rection of the wind is taken at several stations simultane-
ously and by the anemometer its intensity is given. These
data give us the means of drawing on a chart suitably pre-
pared the lines of flow at the given time of day and the
curves showing the points of equal-intensity of the wind
velocity. Of course, the velocity is usually only the hori-
zontal velocity and the vertical velocity must be inferred.

One of the items needed in meteorological and other
studies is the amount of material transported. If the spe-
cific momentum in a horizontal direction is cpr, and lines

20

298 VECTOR CALCULUS

of flow be drawn, then for a vertical height dz and a width
between lines of flow equal to dn, we will have the trans-
port equal to Tp'dndz. Since, however, we have for prac-
tical purposes dz = — dp, we can write this in the form

transport = Tp'dn(— dp).

In order to do this graphically we first draw the lines of
flow and the intensity curves. An arbitrary outer bound-
ary curve is then divided into intervals of arc such that
the projection of an interval perpendicular to the nearest
lines of flow multiplied by the value of Tp' is a constant.
Through these points a new set of lines of flow is constructed.
The transport between these lines is then known horizon-
tally for a constant pressure drop, by drawing the intensity
curves that represent Tp'dn, and if these are at unit values
of the transport, they will divide th£ lines of flow into quad-
rilaterals such that the amount of air transported horizon-
tally decreases or increases by units, and thus the vertical
transport must respectively increase or decrease by units,
through a sheet whose upper and lower surfaces have pies-
sure difference equal to dp = — 1. Towards a center of
convergence the lines of flow approach indefinitely close.
dn decreases and it is clear that the vertical transport up-
ward increases. There may be small areas of descending
motion, however, even near such centers. In this manner
we may arrive at a conception of the actual movement of
the air.

Since the specific momentum is solenoidal, we can as-
certain its rate of change vertically from horizontal data.
For

0 = SVcp' = — dZ/dz + horizontal convergence,
or

dZ/dz = horizontal convergence of specific momentum.

HYDRODYNAMICS 299

Substituting the value of dz, we have

dZ/ (— dp) = horizontal convergence of velocity,
dZ/dp = dTP'lds+ Tp'b.

where ds runs along the lines of flow, and 5 is the diver-
gence per unit ds of two lines of width apart equal to 1.

These considerations enable us to arrive at the complete
kinematic diagnosis of the condition of the air. On this is
based the prognostications.

11. When the density c is a function of the pressure p,
and the forces and the velocities can be expressed as gradi-
ents, then'we have a very simple general case. Thus let

c = f(p), i = V«(p, 0i p' = Vv(p, t),

and set

Q = u — fa&p, then VQ = £ — aVp,

the equations of motion are

dp'/dt + 0(p') = VQ, or since p' = Vv,
V[dv/dt + iT2Vv- Q] = 0.

Hence the expression in brackets is independent of p and
depends only on t and we have

dv/dt+iFW- Q = h(t).

We could, however, have used for v any function differing
from v only by a function of t, thus we may absorb the func-
tion of the right into v and set the right side equal to zero.
We thus have the equations of motion

dv/dt + JPVfl - Q = 0, dc/dt - SV(cVv) = 0,
c = /(p).

From these we have v, c, p in terms of p and /.

12. In the case of a permanent motion, the tubes of flow
are permanent. If we can set £ = Vw(p), then we place

300 VECTOR CALCULUS

Q = u — fadp, and noticing that p' and Q do not depend
on t, we have

Sp'V-p' = - VQ.

If we operate by — Sdp = — S(dsUp'), we have
(kSUp'Tp'VTp' on the left, since Sp'V-Up' = 0. Hence
from this equation we have at once

- SdpGTV - Q) = 0.

Hence along a tube of flow of infinitesimal cross-section

tiy-4-a

This is called Bernoulli's theorem. C is a function of the
two parameters that determine the infinitesimal line of
flow. Hence along the same tube of flow

J(IV - TW) = Q ~ Qo = u - u0 - fp* adp.

In the case of a liquid a is constant and we can integrate
at once, giving

}ZV- u+ap= C.

From this we can find the velocity when the pressure is
given or the pressure when the velocity is given. Since
the pressure must be positive, it is evident that the velocity
square ^ 2{u + C), or else the liquid will separate. This
fact is made use of in certain air pumps. In the case of no
force but gravity we have u = gz,

iTV-g*+ap= C.

This is the fundamental equation of hydraulics. We can-
not enter upon the further consideration of it here.

Vortices.
13. In the case of p' = Vv it is evident that VVpf = 0.
When this vector, or the vector e (§9) does not vanish,

HYDRODYNAMICS 301

there is not a velocity potential and vortices are said to
exist in the fluid. It is obvious that if a particle of the
fluid be considered to change its shape as it moves, then e
is the instantaneous velocity of rotation. At any instant
all the vortices will form a vector field whose lines have the
differential equation

VdpWp' = 0 = SdpV' p - VSp'dp;
that is,

Q'dp = dp', or 0'p' = dp'jdt,

from which

p' . */><%'.

These vector lines are called the vortex lines of the fluid.
Occasionally the vortex lines may be closed, but as a rule
the solutions of such a differential equation as the above
do not form closed lines, in which case they may terminate
on the walls of the containing vessel, or they may wind
about indefinitely. The integral of this equation will
usually contain t, and the vortices then vary with the time,
but in a stationary motion they will depend only upon the
point under consideration.

14. The equations of motion may be expressed in terms
of the vortex as follows, since

we have
and thus

Vp'VVp' ' = Sp'V-p'-iVp'2,.

Sp'V-p' =2Vp'e + iVp'\

aVp = i - dp' Idt + JVp/2 + 2Vp'e.

15. When now £ = \/u{p, t), and c = f(p), we set
P = fadp, giving VP = aVp, and thence

VP = Vu - dp' Idt + JVp/2 - 2Vep'.

302 VECTOR CALCULUS

Or, if we set II = u-\- Jp'2 — P, we have

dp'/dt + 2VeP' = VII.

Operate on this with V-V(), and since VV dp'/dt
= 2de/dt, and WVep' = SeV -p' - eSVp' - Sp'V-e, de/dt
— Sp'V-e = de/dt, SVp' by the continuity equation is
equal to c~ldc/dt = — a~lda/dt, we have

d(ae)/dt = - S(ae)V-p' = 6(ae).

This equation is due to Helmholtz.

If we remember the Lagrangian variables, it is clear that
6 is a function of the initial vector p0 and of t, hence the
integral of this equation will take the form

ae = efm'a,eQ = e' ~ s^^'dta0e0 = ^(t)a0e0.

But the operator is proved below to be equal to <p itself,
so that when £ = Vu,

ae = a0Se0Vo-p = + ao<p€0,

or finally we have, if we follow the stream line of a particle,
which was implied in the integration above, Cauchy's form
of the integral

(a/a0)e = — *Se0Vo'jP,

where p is a function of p0 and t. It is evident now if
for any particle e is ever zero, that is, e0 = 0, that always
e = 0. This is equivalent to Lagrange's theorem that if
for any group of particles of the fluid we have a velocity
potential, then that group will always possess a velocity
potential. (It is to be noted that velocity potential and
vortex are phenomena that belong to the particles and the
stream lines, and not to the points of space and the lines
of flow.) It must be remembered too, that this result
was on the supposition that the density was a function of

HYDRODYNAMICS 303

the pressure alone, and that the external forces £ were
conservative.

16. We may deduce the equation above as follows, which
reproduces in vector form the essential features of Cauchy's
demonstration. (Appell, Traite de Mec. Ill, p. 332.)

Let dp/dt = a, and Q = u — fadp, then, remembering
that Q is a function of p and t, and p is a function of p0
and /,

da/dt = VQ(p, t).

Also VoQ(po, t) = - VoSpVQ = — VoSpda/dt, where Vo
operates on p only; or we can write

VoQ = <p' da/dt.

Hence, operating with FVo( ), we have V\7o(pf da/dt = 0 =
d/dt(VVo<p'a). Thus the parenthesis equals its initial
value, that is, since the initial value of cp'a is a0, and since
Vo = <p'V,

VVo<p'<r = 2e0 = V<p'Vv'<r = mz{<p)<p~lVS7a = 2m3(p~1e.

Thus we have at once m3e = (pe0. This is the same as the
other form, since ra3 = a/aQ. This equation shows the
kinematical character of e, and that no forces can set up e
or destroy it.

17. The circulation at a given instant of the velocity
along any loop is

I = - fSdpp'.

The time derivative of this is dl/dt = tf^SdpS/Sp'p'
- Sdpp") = £(- SdpW tip'2 - Q] ). But this is an inte-
gral of an exact differential and vanishes. Hence if the
forces are conservative and the density depends on the
pressure, the circulation around any path does not change
as the particles of the path describe their stream lines. The

304 VECTOR CALCULUS

circulation is an integral invariant. This theorem is due to
Lagrange. If we express the circulation in the form

I - ' - ffSdvVp' = - 2ffSdpe,

we see that the circulation is twice the flux of the vortex
through the loop. Hence as the circulation is constant,
the flux of the vortex through the surface does not vary
in time, if the surface is bounded by the stream loop. The
flux of the vortex through any loop at a given instant is
the vortex strength of the surface enclosed by the loop.
If a closed surface is drawn in the fluid, the flux through it
is zero, since the vortex is a solenoidal vector.

18. If we take as our closed surface a space bounded by a
vortex tube and two sections of the tube, since the surface
integral over the walls of the tube is zero, it follows that
the flux of the vortex through one section inwards equals
that over the other section outwards. Combining these
theorems, it is evident that the vortex strength, or wr-
ticity, of a vortex tube is constant. Thus the collection of
particles that make up the vortex tube is invariant in time.
In a perfect fluid a vortex tube is indestructible, and one
could not be generated.

19. It is evident from what precedes that a vortex tube
cannot terminate in the fluid but must end either at a wall
or a surface of discontinuity, or be a closed tube with or
without knots, or it may wind around infinitely in the fluid.

If a vortex tube is taken with infinitesimal cross-section,
it is called a vortex filament.

20. We consider next the problem of determining the
velocity when the vortex is known. That is, given e, to
find a = p'. We consider first the case of an incompressible
fluid, in which the velocity is solenoidal, that is, SVcr = 0.
This with the equations at the boundaries gives us the

HYDRODYNAMICS 305

following problem : to find a when 2e = FVo", SVer = 0,
SUva = 0 at the boundaries, or if infinite aa = 0. This
problem has a unique solution, if the containing vessel is
simply connected. We cannot enter extensively into it,
for it involves the theory of potential functions, and may
be reduced to integral equations. However, since SVv = 0,
we may set a = VVr, where *$Vr = 0, whence

V2r = 2e,

and we may suppose r is known, in the form

T = h7ffSfe/T(p- Po)dv.

If we operate upon this by FV( ), we find a formula for a,

a = H,2ir-fffVe(p - p0)/T\p - Po)dv.

As we see, this formula is capable of being stated thus:
the velocity is connected with its vortex in the same way
as a magnetic field is connected with the electric current
density that produces it, the vortex filament taking the
place of the cm rent, the strength of current being Tej2ir,
and the elements of length of the tube acting like the ele-
ments of current. This solution holds throughout the
entire fluid, even at points outside the space that is actually
in motion with a vortex.

Since the equation of the surface of the tube can be
written in the form

F(P, t) = 0,

this surface will move in time. Its velocity of displace-
ment is defined like that of any discontinuity, as
UvFdF/dt. On one side the velocity is irrotational, on
the other it is vortical. On the irrotational side we have
the velocity of the form a — V?, and we must have on

306 VECTOR CALCULUS

that side the same velocity of displacement in the form

UpSUpVP.

The energy involved in a vortex on account of the velocity
in the particles is

K * - \cfffp'2dv
= " hcfffSp'Vrdv
= ¥fff [SV(p'r) - 2Sre]dv
= hcffSdvp'r - cfffSredv
= — cj J 'fSredv over all space
= c/2T.SffSSSSee'lT(p - p0)dvdv'.

This is the same formula as that of the energy of two cur-
rents. In the expression every filament must be considered
with regard to every other filament and itself.

Examples. (1). Let there be first a straight voitex fila-
ment terminating at the top and bottom of the fluid. Let
all the motion be parallel to the horizontal bottom. Then

Sya = 0, Vye = 0, de/dt = 0.

We have then

a = VyVw, 2e = — yV2w = 2zy,
say,

w = — 7r lffz log rdA.

For a single vortex filament of cross-section dA and strength
k = zdA, we have

iv = — k/w log r = — kjir log V (#2 + 2/2)
a= Vy(p- po)IT>(p- p0).k/T,

where p is measured parallel to the bottom.

The velocity is tangent to the circles of motion and in-
versely as the distance from the vortex filament. The
motion is irrotational save at the filament itself.

HYDRODYNAMICS 307

For the effect of vortices upon each other, and their
relative motions, see Webster, Dynamics, p. 518 et seq.

(2). For the case of a vortex ring or a number of vortex
rings with the same axis, see Appell, Traite, vol. Ill, p. 431
et seq.

21. In the more general case in which the fluid is com-
pressible we must resort to the theorem that any vector
can be decomposed into a solenoidal part and a lamellar
part and these may then be found. The extra term in the
electromagnetic analogy would then be due to a perma-
nent distribution of magnetism as well as that arising fiom
the current.

EXERCISES

1. If Sea = 0, then it is necessary and sufficient that a = M\/P,
M being a function of p.

2. Discuss the case Vae = 0. Beltrami, Rend. R. 1st. Lomb. (2)
22, fasc. 2.

3. Discuss Clebsch's transformation in which we decompose <r thus,
o- = Vm + lVV. Show that the vortex lines are the intersections of
the surfaces I and v, and that the lines of flow form with the vortex lines
an orthogonal system only when the surfaces I, u, v are triply orthog-
onal.

4. Discuss the problem of sources and sinks.

5. Consider the problem of multiply-connected surfaces, containing
fluids.

22. It will be remembered that Helmholtz's theorem
was for the case in which the impressed forces had a poten-
tial and the density was a function of the pressure. In
this case we will have the equation

da/dt + 2Vea = { - aVp + JVtf2.

Operate by |FV( ) and notice that

de/dt - eSVa - SaV-e = a -ld(ae)/dt,

whence we have the generalized form

a-ld(ae)ldt + SeV -<r = iVV£ - fFVaVp.

308 VECTOR CALCULUS

If now at the instant t0 the particle does not rotate and if a
is a function of p alone, then at this instant de/dt = JFV£,
and the paiticle will acquire an instantaneous increase of
its zero vortex equal to the vortex of the impressed force.
That is, £ must be peimanently equal to zero if there is to
be no rotation at any time.

If FV£ = 0 but a is not a function of p alone, then we
have

a-1d(ae)/dt + SeV -<r = - §WaVp.

The right side is a vector in the direction of the intersection
of the isobaric and the isosteric surfaces. Now if we take
an infinitesimal length along the vortex tube, I, the cross-
section being A, the vorticity is ATe = m, the mass is
cAl = constant = M. Then we have, since ae = AlejM
= mlUejM,

- SeV-<r = md(lUe)dtaM « - ~ fUeV* - ^^-elf

I I at

a-1d(ae)/dt+ SeV -a =

dmldt-lUe/aM'+ md{We)la Mdt - md(lUe)/dtaM =

dm/dt-lUe/aM = Ve-dTe/dt =

±| number of tubes.

Hence the moment m of the vortex will usually change
with the time unless the surfaces coincide. Thus a rotat-
ing particle may gain or lose in vorticity. If then the
isobaric and isosteric surfaces under the influence of heat
conditions intersect, vortices will be created along the lines
of intersections of the surfaces and these will persist until
the surfaces intersect again, save so far as viscosity
interfeies.

23. Finally we consider the conditions that must be
put upon surfaces of discontinuity, in this case of the first
order in <r, that is, a wave of acceleration.

HYDRODYNAMICS 309

Let c be a function of p only. Then

a\/p = dp/dc \7log c, and the equation of motion becomes
p" = J — dp/dc • V log c.

Let the equation of the surface of discontinuity be f(p0, t)
= 0, the normal v. Let £, a, p, and c be continuous as
well as dp/dc, but p" = a' be discontinuous at the suiface.
Then on the two sides of the surface we have the jump,

by p. 263,

\p"\ = - dp/dc[V log c],
or

G2ix= dp/dc -UVfSfiUVf.

It follows, therefore, that we must have V/iUVf = 0 and
G = V (dp/dc), or else we have G = 0 and SnUVf = 0.
In the first case the discontinuity is longitudinal, in the
second transversal. This is Hugoniot's theorem. In full
it is:

In a compressible but non- viscous fluid there are possible
only two waves of discontinuity of the second order; a
longitudinal wave propagated with a velocity equal to
V (dp/dc), and a transversal wave which is not propagated
at all.

The formula for the velocity in the first case is due to
Laplace. Also we have for the longitudinal waves [&Vo"]
= — GSfxUVf, for transversal waves equal to zero. On
the other hand, for longitudinal waves, [FVo\| = 0, for
transversal, = GVUVf^.

310 VECTOR CALCULUS

REFERENCES.

1. Mathematische Schriften (Ed. Gerhart). Berlin, 1850. Bd. II,
Abt. 1, p. 20.

2. On a new species of imaginary quantities connected with a theory
of quaternions. Proc. Royal Irish Academy, 2 (1843), pp. 424-434.

3. Die lineale Ausdehnungslehre. Leipzig, 1844.

4. Gow: History of Greek Mathematics, p. 78.

5. Ars Magna, Nuremberg, 1545, Chap. 37; Opera 4, Lyon, 1663, p.
286.

6. Algebra. Bologna, 1572, pp. 293-4.

7. Om Directiones analytiske Betejning. Read 1797. Nye Samm-
lung af det kongelige Danske Videnskabernes Selskabs Skrifter, (2)
5 (1799), pp. 469-518. Trans. 1897. Essai sur la representation de
la direction, Copenhagen.

8. Essai sur une maniere de repr6senter les quantites imaginaires
dans les constructions g6ometriques. Paris, 1806.

9. Theoria residuorum biquadraticum, commentation secunde. 1831.

10. Annales Math, pures et appliqu6es. 4 (1814-4), p. 231.

11. Theory of algebraic couples, etc. Trans. Royal Irish Acad., 17
(1837), p. 293.

12. Ueber Functionen von Vectorgrossen welche selbst wieder Vector-
grossen sind. Math. Annalen, 43 (1893), pp. 197-215.

13. Grundlagen der Vektor-und Affinor- Analysis. Leipzig, 1914.
\\. Lectures on Quaternions. Preface. Dublin, 1853.

15. Note on William R. Hamilton's place in the history of abstract
group theory. Bibliotheca Mathematica, (3) 11 (1911), pp. 314-5.

16. Leipzig, 1827.

17. Leipzig.

18. Elements of Vector Analysis (1881-4), New Haven. Vol. 2,
Scientific Papers.

INDEX.

Acceleration 27

Action : 14, 28

Activity 15, 129, 142

Activity-density 15, 131

Algebraic couple 4, 65

Algebraic multiplication 9

Alternating current 71

Ampere 30

Anticyclone 47

Area 142

Areal axis 198

Argand 4

Ausdehnungslehre 3, 9

Average velocity 57

Axial vector 30

Barycentric calculus 8

Bigelow 50, 60

Biquaternions 3, 126

Biradials 94

Bivector 29

Bjerknes 48, 57, 59, 290

Cailler 2

Cardan 3

Center (singularity) 44

Center of isogons 48

Change of basis 54

Characteristic equation 125

Characteristic equation of

dyadic 221

Chi of dyadic 235

Christoffel's conditions 266

Circuital derivative 167

Circular multiplication 9

Circulation 78, 129

Clifford 3, 90

Combebiac 3

Complex numbers 63

Congruences 51, 138

Conjugate 66

Conjugate function 5

Continuous group 195

Continuous plane media 87

Convergence 177

Coulomb 13

Couple 139

Crystals 109

Cubic dilatation 258

Curl 76,82, 184

Curl of field 77

Curvature 148, 152

Curves 148

Cycle 30, 37

Cyclone 47

Derivative dyad 242

Developables 150

Dickson 105

Differential of p . . . ' 145

Differential of q 155, 159

Differential of vector 55

Differentiator 248

Directional derivative 166

Discharge 130

Discontinuities 261

Dissipation (plane) 84

Dissipation, dispersion 180

Divergence 76, 82

Divergence of field 77

Dyadic 2, 11, 218

Dyadic field 246

Dyname . 2

Dyne 29

Electric current 30

Electric density current 30

Electric induction 32

Electric intensity 31, 139

Energy 14

Energy current 30

Energy-density 15, 131

Energy-density current 30

Energy flux 142

Equation of continuity 87

Equipollences 71

Equipotential 15

Erg 14

Euler 107

Exact differential 190

Exterior multiplication 9

Extremals 160

Eye of cyclone 47

311

:;il>

VECTOR CALCULUS

Farad 32, 73

Faux 37, 38

Faux-focus 44

Feuille 30

Feuillets 2

Field 13

Flow 142

Flux 29, 130, 142

Flux density 29

Focus 41

Force 29

Force density 28, 141

Force function 18

Franklin 90

Free vector 8, 25

Frenet-vSerret formulae 148

Functions of dyadic 238

Function of flow 88

Functions of quaternions. ... 121

Gas defined 87

Gauss 4/

Gauss (magnetic unit). . . .32, 130

Gaussian operator 108

General equation of dyadic . . 220

Geometric curl 76

Geometric divergence 76

Geometric loci 133

Geometric vector 1

Geometry of lines 2

Gibbs 2, 11, 215

Gilbert 32, 130, 143

Glissant 26

Gradient 16, 163

Gram 15

Grassmann 2, 3, 9

Green's Theorem 205

Groups 8

Guiot 138

Hamilton 2, 3, 4,65,95

Harmonics 84, 169

Heaviside 31

Henry (electric unit) 32, 73

Hertzian vectors 33

Hitchcock 49

Hodograph 27

Hypernumber 3, 94

Imaginary 65

Impedance 73

Inductance. 73

Inductivity 32

Integral of vector 56

Integrating factor 191

Integration by parts 198

Interior multiplication 10

Invariant line 219

Irrotational 88

Isobaric 15,288

Isogons 34

Isohydric 15

Isopycnic 15, 288

Isosteric 15, 288

Isothermal 15

Joly 138, 147

Joule 14

Joule-second 14

Kinematic compatibility .... 266

Kirchoff's laws .' 73

Koenig 198, 205

Laisant 71

Lamellae 15

Lamellar field 84, 181

Laplace's equation 214

Latent equation 220

Laws of quaternions 103

Leibniz 3

Level 15

Line (electric unit) 32, 130

Lineal multiplication 9

Linear associative algebra ... 3

Linear vector function 218

Line of centers 46

Line of convergence 47

Line of divergence 47

Line of fauces 46

Line of foci 46

Line of nodes 45

Lines as levels 80

Liquid defined 87

MacMahon 75

Magnetic current 31

Magnetic density current 31

Magnetic induction 32

Magnetic intensity 32, 139

Mass 15

Matrix unity 65

Maxwell 13

McAulay 3

Mobius 8

Modulus 66

Moment 138

Moment of momentum 139

INDEX

313

Momentum 28

Momentum density 28

Momentum of field 141

Monodromic 14

Monogenic 89

Moving electric field 140

Moving magnetic field 140

Multenions 3

Multiple 6

Mutation 108

Nabla as complex number. . . 82

Nabla in plane 80

Nabla in space 162

Neutral point 47

Node 37,38

Node of isogons 48

Non-degenerate equations . . . 225

Norm 66

Notations

One vector 12

Scalar 127

Two vectors 136

Derivative of vectors 165

Divergence, vortex, deriva-
tive dyads 179

Dyadics 248

Ohm (electric unit) 73

Orthogonal dyadic 241

Orthogonal transformation . . 55

Peirce, Benjamin 3

Peirce, B. O 85

Permittance 73

Permittivity 32

Phase angle 71

Plane fields.. 84

Poincare 36, 46

Polar vector 30

Polydromic 14

Potential ,. .. 15, 17

Progressive multiplication ... 10

Power 76

Poynting vector 141

Pressure 142

Product of quaternions 98

Product of several quater-
nions 113

Product of vectors 101

Quantum 14

Quaternions 2, 3, 6, 7, 95

Radial 26

Radius vector 26

Ratio of vectors 62

Reactance 73

Real 65

Reflections \ 108

Refraction . . 112

Regressive multiplication. ... 10

Relative derivative 18

Right versor 96

Rotations 108

Rotatory deviation 175

Saint Venant's equations. . . . 260

Sandstrom 35, 49

Saussure 2

Scalar 13

Scalar invariants 220, 239

Scalar of q 96

Schouten 7

Science of extension 2

Self transverse 234

Servois 4

Shear 256

Similitude 242

Singularities of vector lines . . 244

Singular lines 45

Solenoidal field 84, 181

Solid angles 117

Solution of equations 123

Solution of differential equa-
tions 195

Solution of linear equation. . . 229

Specific momentum 28

Spherical astronomy 110

Squirt 90

Steinmetz 68, 71

Stoke's theorem 200

Strain 253

Strength of source or sink ... 90

Stress 143,269

Study . . . 2

Sum of quaternions 96

Surfaces 151

Symmetric multiplication ... 9

Tensor 65

Tensor of q 96

Torque 140

Tortuosity 149

Trajectories 150

Transport 130,298

Transverse dyadic 231

Triplex 25

314

VECTOR CALCULUS

Triquaternions 3

Trirectangular biradials 100

Unit tube 18

Vacuity 220

Vanishing invariants 240

Variable trihedral 172

Vector 1

Vector calculus 1, 25

Vector field 23, 26

Vector lines 33

Vector of q 96

Vector potential 33, 93, 181

Vector surfaces 34

Vector tubes 34

Velocity 27

Velocity potential 18

Versor 65

Versor of q 96

Virial 129

Volt 31, 130, 143

Vortex 92, 187, 187

Vorticity 247,304

Waterspouts 50

Watt 15

Weber 14

Wessel 4

Whirl 90

Zero roots of linear equations. 230

foist r Ot— C/ p^V A^y

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